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 8 Grade
Intensive Math
th
Name _______________________________________ Date __________________ Class __________________
LESSON
6-5
Practice A
Percent of Increase and Decrease
State whether each change represents an increase or decrease.
1. from 10 to 15
_______________________
2. from 16 to 12
3. from 8 to 14
________________________
________________________
Find each percent of increase or decrease to the nearest
percent.
4. from 2 to 5
_______________________
7. from 8 to 5.6
_______________________
10. from 17 to 21
_______________________
13. from 7 to 11
_______________________
5. from 10 to 6
6. from 12 to 18
________________________
8. from 15 to 8
________________________
9. from 21 to 15
________________________
11. from 10 to 2
________________________
12. from 4 to 9
________________________
14. from 3 to 9
________________________
15. from 12 to 5
________________________
________________________
16. World Toys buys bicycles for $38 and sells them for $95.
What is the percent of increase in the price?
_________________
17. Jack bought a stereo on sale for $231. The original price
was $385. What was the percent of decrease in price?
_________________
18. Adams Clothing Store buys coats for $50 and sells them
for $80. What percent of increase is this?
_________________
19. Asabi’s average in math for the first quarter of the school
year was 75. His second quarter average was 81. What was
the percent of increase in Asabi’s grade?
_________________
20. A shoe store is selling athletic shoes at 30% off the regular
price. If the regular price of a pair of athletic shoes is $45,
what is the sale price?
_________________
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362
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-5
Reading Strategies
Compare and Contrast
Percent can be used to describe change. It is shown as a ratio.
amount of change
percent of change =
original amount
Compare the two lists. Change can either increase or decrease.
Decrease
Increase
A collector sold 15 CDs.
Then she sold 25 more CDs.
Ben had a collection of 60 CDs.
Now he has only 45 CDs.
Sales went up, so the ratio will show a
percent of increase.
The CD collection went down, so the
ratio will show a percent of decrease.
Change: 25 15 = 10 more CDs
10
Percent of change =
25
Change fraction to percent: 40%
Change: 60 45 = 15 fewer CDs
15
Percent of change =
60
Change fraction to percent: 25%
1. Compare percent of increase with percent of decrease. How are they the same?
________________________________________________________________________________________
2. Write the ratio that stands for percent of change.
________________________________________________________________________________________
Write percent of increase or percent of decrease to describe
each situation.
3. Sophie had $70 saved. She withdrew $15 from her savings.
________________________________________________________________________________________
4. Kate bought $50 worth of groceries. Then she bought $20 more.
________________________________________________________________________________________
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369
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-5
Review for Mastery
Percent of Increase and Decrease
To find the percent of increase:
• Find the amount of increase by subtracting the lesser number from the greater.
• Write a fraction: percent of increase =
amount of increase
original amount
• If possible, simplify the fraction.
• Rewrite the fraction as a percent.
The temperature increased from 60°F to 75°F.
Find the percent of increase.
75° 60°
15°
1
percent of increase =
=
=
= 25%
60°
60°
4
Complete to find each percent of increase.
1. Membership
2. Savings
increased from
increased from
80 to 100.
$500 to $750.
_________________
_________________
= __________
= __________
80
= __________
= __________ %
500
= __________
= __________ %
Find the amount of increase.
percent of increase =
amount of increase
original amount
Change the fraction to a percent.
3. Price increased
from $20 to $23.
Find the amount of increase.
_________________
percent of increase =
20
20
= 20
= _________________ %
amount of increase
original amount
Change the fraction to a percent.
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365
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-5
Review for Mastery
Percent of Increase and Decrease (continued)
To find the percent of decrease:
• Find the amount of decrease by subtracting the lesser number from the greater.
amount of decrease
original amount
• Write a fraction: percent of decrease =
• If possible, simplify the fraction.
• Rewrite the fraction as a percent.
Carl’s weight decreased from 175 lb to 150 lb.
Find the percent of decrease.
0.143
175 150
25
1
percent of decrease =
=
=
= 7 1.000 = 14.3%
175
175
7
Complete to find each percent of decrease.
4. Enrollment
5. Temperature
decreased from
decreased from
1000 to 950.
75°F to 60°F
_________________
_________________
= __________
= __________
1000
=
100
= __________ %
75
=
15
=
Find the amount of decrease.
percent of decrease =
100
= __________ %
amount of decrease
original amount
Change the fraction to a percent.
6. Sale price
decreased from
$22 to $17.
Find the amount of decrease.
_________________
percent of decrease =
22
22
= 22
= _________________ %
amount of decrease
original amount
Change the fraction to a percent.
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366
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
6-5 Percent of Increase and Decrease
Problem 1
36 to 45 . . . is this an increase or a decrease?
bigger amount
Decrease: bigger amount to smaller amount
Increase: smaller amount to
Always subtract
smaller number
from greater number.
Amount of Change 45 36 9
amount of change
Multiply by 100.
9
Percent Change 3
0.25
6
original amount
25%
The amount increased
by 25% because you
went from a smaller to
a greater amount.
Problem 2
Original Price
$750
Sale Percentage
Discount
35%
d
BIG
SALE
C
OM
$750
S
ITOR
P U T E R M ON !
To d a y O n l
y!!
750 • 0.35 d
262.50 d
Sav
e
35 %
So, Anthony
saved $262.50.
How much does Anthony have to pay for his monitor?
Original Price
Amount Saved
Sale Price
$750
$262.50
s
750 262.50 s
487.50 s
So, Anthony bought his
monitor on sale for $487.50.
Think and Discuss
1. What words in a mathematical sentence signify a decrease?
2. What words in a mathematical sentence signify an increase?
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84
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class__________________
LESSON
6-5
Problem Solving
Percent of Increase and Decrease
Use the table below. Write the correct answer.
1. What is the percent of increase
in the population of Las Vegas,
NV from 1990 to 2000? Round
to the nearest tenth of a percent.
Solution:
Fastest Growing Metropolitan Areas,
1990–2000
Metropolitan
Area
Population
1990
2000
Increase
1,563,282 852,737
= 0.833
852,737
Las Vegas,
NV
852,737
1,563,282
0.833 100 = 83.3%
Naples, FL
152,099
251,377
Yuma, AZ
106,895
2. What is the percent of increase
in the population of Naples,
FL from 1990 to 2000?
Round to the nearest tenth
of a percent.
152,099
Percent
of
49.7%
3. What was the 2000 population of
Yuma, AZ to the nearest whole
number?
= _______ 100 = _______%
________________________________________
For Exercises 4–6, round to the nearest tenth. Choose the letter
for the best answer.
5. In 1967, a 30-second Super Bowl
commercial cost $42,000. In 2000,
a 30-second commercial cost
$1,900,000. What was the percent
increase in the cost?
4. The amount of money spent on
advertising costs in 2000 was 4.4%
lower than in 1999. If the 1999
spending was $1,812.3 million, what
was the 2000 spending?
A $79.7 million
A 44.2%
B $1,732.6 million
B 442.4%
C $1,892 million
C 4,423.8%
6. In 1896 Thomas Burke of the U.S.
won the 100-meter dash at the
Summer Olympics with a time of
12.00 seconds. In 2004, Justin
Gatlin of the U.S. won with a time of
9.85 seconds. What was the percent
decrease in the winning time?
A 2.15%
B 17.9%
C 21.8%
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104
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-6
Practice A
Applications of Percents
Let c = the commision amount and write an equation to find the
commission for the following. Do not solve.
1. 10% commission on $4000
2. 6% commission on $8450
_______________________________________
________________________________________
3. 8% commission on $3575
4. 12% commission on $12,750
_______________________________________
________________________________________
5. 5.5% commission on $60,000
6. 6
_______________________________________
1
% commission on $85,900
4
________________________________________
Write a proportion to represent the following. Do not solve.
7. What percent of 14 is 7?
8. 7 is what percent of 25?
_______________________________________
________________________________________
9. What number is 12.5% of 16?
10. 21 is 35% of what number?
_______________________________________
________________________________________
Solve.
11. 45 is 25% of what number?
12. What percent of 288 is 36?
_______________________________________
________________________________________
13. A financial investment broker earns 4% on each customer
dollar invested. If the broker invests $50,000, what is
the commission on the investment?
______________
14. Sharlene bought 4 CDs at the music store. Each cost $14.95.
She was charged 5% sales tax on her purchase. What was
the total cost of her purchase?
______________
15. Isaac earned $1,800 last month. He put $270 into savings.
What percent of his earnings did Isaac put in savings?
______________
16. Edel works for a company that pays a 15% commission on
her total sales. If she wants to earn $450 in commissions,
how much do her total sales have to be?
______________
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371
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-6
Reading Strategies
Focus On Vocabulary
A commission is a percent of money a person is paid for making a
sale. Many salespeople receive a commission on the amount
they sell.
The commission rate is the percent paid on a sale. A salesperson
might receive a 5% commission in addition to his salary. The
commission rate is 5%.
The formula for finding out how much a salesperson earns based on
the commission rate and the amount of sales is:
commission rate • sales = amount of commission
Sales tax is added to the price of an item or service. Sales tax is a
percent of the purchase price. A sales tax of 6.5% means that all
taxable items will have an additional 6.5% added to the total cost.
sales tax rate • sale price = sales tax
sale price + sales tax = total sale
The total sale price is computed by adding the sales tax to the cost
of all the items purchased.
Write commission, commission rate, sales tax, or total sale to
describe each situation.
1. $5.45 was added to the price of the shoes Jill bought.
________________________________________________________________________________________
2. The man who sold your family a car receives $500 for the sale.
________________________________________________________________________________________
3. Mr. Adams makes a 4% commission on each house he sells.
________________________________________________________________________________________
4. Caroline spent $37.43 for two shirts plus tax.
________________________________________________________________________________________
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378
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-6
Review for Mastery
Applications of Percents
Salespeople often earn a commission, a percent of their total sales.
Find the commission on a real-estate sale of $125,000
if the commission rate is 4%.
Write the percent as a decimal and multiply.
commission rate amount of sale = amount of commission
0.04 $125,000
= $5000
If, in addition to the commission, the salesperson earns a
salary of $1000, what is the total pay?
commission + salary = total pay
$5000 + $1000 = $6000
Complete to find each total monthly pay.
1. total monthly sales = $170,000; commission rate = 3%; salary = $1500
amount of commission = 0.03 $________________ = $________________
total pay = $________________ + $1500 = $________________
2. total monthly sales = $16,000; commission rate = 5.5%; salary = $1750
amount of commission = ________________ $________________ = $________________
total pay = $________________ + $________________ = $________________
A tax is a charge, usually a percentage, generally imposed by a government.
If the sales tax rate is 7%, find the tax on a sale of $9.49.
Write the tax rate as a decimal and multiply.
tax rate amount of sale = amount of tax
0.07
$9.49
= $0.6643 $0.66
Complete to find each amount of sales tax.
3. item price = $5.19; sales tax rate = 6%
amount of sales tax = 0.06 $_____________ = $_____________ $_____________
4. item price = $250; sales tax rate = 6.75%
amount of sales tax = _____________ $_____________ = $_____________ $_____________
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374
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-6
Review for Mastery
Applications of Percents (continued)
Use a proportion to find what percent of a person’s income
goes to a specific expense.
Heather earned $3,200 last month. She paid $448 for
transportation. To find the percent of her earnings that she
put towards transportation, write a proportion.
Think: What percent of 3200 is 448?
n
100
=
448
3200
Set up a proportion.
Think:
part
part
=
whole
whole
3200n = 448 100
Find cross products.
3200n = 44,800
Simplify.
3200n
3200 =
44,800
3200
n = 14
Divide both sides by 3200.
Simplify.
Heather put 14% of her earnings towards transportation.
Complete each proportion to find the percent of earnings.
6. Leah earned $1,900 last month.
She paid $304 for utilities. What
percent of her earnings went to
utilities?
5. Wayne earned $3,100 last month.
He paid $837 for food. What percent
of his earnings went to food?
n
=
3100
100
n
304
=
100
3100n = ________ 100
________ n
3100n = ________
3100n
= ________
3100
________
= ________
________
= ________
n = ________
n = ________
________
= ________ 100
of Wayne's earnings
went to food.
________
of Leah’s earnings went to
utilities.
.
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375
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
6-6 Applications of Percents
Problem 1
Just to inform you, I get
a 4% commission if I sell
a car.
For Sale $39,500
Commission
is
Commission Rate
of
Total Sales
c
4%
•
$39,500
c 4% • $39,500
c 0.04 • 39,500
c 1580
Write the equation.
Change the percent to a decimal.
Multiply.
Julie will be paid an additional
$1580 for selling the car.
Problem 2
Find the tax on the sale.
1@ 145.80
2@ 15.99
$145.80
$31.98
Subtotal
$177.78
Tax (7.75%)
t 7.75% • $177.78
t 0.0775 • 177.78
t 13.78
Add the cost of
total purchases
to get a subtotal.
Multiply the subtotal
by the tax rate.
$13.78
So, Meka would pay $13.78 in tax
for her DVD player and DVDs.
Think and Discuss
1. How do you determine how much money Meka had to pay in
total for her DVD player and DVDs in Problem 2?
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86
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-7
Practice A
Simple Interest
Write the formula to compute the missing value. Do not solve.
1. principal = $100
2. principal = $150
rate = 4%
rate = ?
time = 2 years
time = 2 years
interest = ?
interest = $9
_______________________________________
________________________________________
3. principal = $200
4. principal = ?
rate = 5%
rate = 3%
time = ?
time = 4 years
interest = $10
interest = 30
_______________________________________
________________________________________
5. Jules borrowed $500 for 3 years at a simple interest rate of 6%.
How much interest will be due at the end of 3 years? How much
will Jules have to repay?
________________________________________________________________________________________
6. Karin maintained a balance of $250 in her savings account for
8 years. The financial institution paid simple interest of 4%. What
was the amount of interest earned?
________________________________________________________________________________________
Complete the table.
7.
8.
9.
10.
11.
12.
13.
Principal
$300
$450
$500
$700
$750
$800
Rate
3%
4.5%
8%
4%
Time
4 years
3 years
2 years
3 years
2 years
2.5%
Interest
$67.50
$112.50
$108
$90
$100
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380
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-7
Reading Strategies
Focus on Vocabulary
Interest is the amount of money the bank pays you to use your
money, or the amount of money you pay the bank to borrow its
money.
Principal is the amount of money you save or borrow from the bank.
Rate of interest is the percent rate on money you save or borrow.
Time is the number of years the money is saved or borrowed.
Use this information to answer Exercises 1–3:
You put $800 in a savings account at 4% interest and leave it
there for five years.
1. What is the principal?
________________________________________________________________________________________
2. What is the interest rate?
________________________________________________________________________________________
3. What is the amount of time the money will stay in the account?
________________________________________________________________________________________
You can find out how much interest you would earn on that money
by using this formula:
Interest = principal • rate • time
words
I
=
p
•
r
• t
symbols
I
=
$800
• 4% • 5
I
=
$800
• 0.04 • 5
Change % to decimal.
I
=
$160
Multiply to solve.
4. To find out how much interest you will earn by keeping your
money in a bank, what three things do you need to know?
________________________________________________________________________________________
________________________________________________________________________________________
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387
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-7
Review for Mastery
Simple Interest
Interest is money paid on an investment.
A borrower pays the interest. An investor earns the interest.
Simple interest, I, is earned when
an amount of money, the principal P,
is borrowed or invested at a rate of interest r
for a period of time t.
Interest = Principal • Rate • Time
I= P•r•t
Situation 1: Find I given P, r, and t.
Calculate the simple interest on a loan of $3500
for a period of 6 months at a yearly rate of 5%.
5% = 0.05
6 months = 0.5 year
Write the interest rate as a decimal.
Write the time period in terms of years.
I=P•r•t
I = 3500 • 0.05 • 0.5 = $87.50
interest earned
Find the interest in each case.
1. principal P = $5000; time t = 2 years; interest rate r = 6%
I = P • r • t = ____________ • 0.06 • _______ = $ _______
2. principal P = $2500; time t = 3 months; interest rate r = 8%
I = P • r • t = ____________ • ____________ • ____________ = $ _______
Situation 2: Find t given I, P, and r.
I=P•r•t
390 = 3000 • 0.065 • t
390 = 195t
390 195t
=
195
195
2=t
An investment of $3000 at a yearly rate
of 6.5% earned $390 in interest. Find
the period of time for which the money
was invested.
The investment was for 2 years.
Find the time in each case.
3. I = $1120; P = $4000; r = 7%
4. I = $812.50; P = $5000; r = 6.5%
I=P•r•t
1120 = _______ • 0.07 • t
812.50 = _______ • _______ • t
1120 = _______ t
812.50 = _______ t
t
=
______________
_______
I=P•r•t
t
=
______________
______________
years = t
_______
______________
years = t
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383
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
6-7
Review for Mastery
Simple Interest (continued)
Situation 3: Find r given I, P, and t.
$2500 was invested for 3 years
and earned $450 in interest.
Find the rate of interest.
The interest rate was 6%.
I=P•r•t
450 = 2500 • r • 3
450 = 7500r
450
7500r
=
7500
7500
0.06 = r
Find the interest rate in each case.
5. I = $1200; P = $6000; t = 4 years
6. I = $325; P = $2000; t = 2.5 years
I=P•r•t
I=P•r•t
1200 = _______ • r • 4
325 = _______ • r • _______
1200 = _______ r
325 = _______ r
=
_______
r
r
=
=r
__________
The interest rate was _______ %
=r
The interest rate was
The total amount A of money in an account
after interest has been earned, is the sum
of the principal P and the interest I.
_______
%.
Amount = Principal + Interest
A= P+ I
Find the amount of money in the account after $3500
has been invested for 3 years at a yearly rate of 6%.
First, find the interest earned.
I=P•r•t
I = 3500 • 0.06 • 3 = $630
interest earned
Then, add the interest to the principal.
3500 + 630 = 4130
So, the total amount in the account after 3 years is $4130.
Find the total amount in the account.
7. principal P = $4500; time t = 2.5 years; interest rate r = 5.5%
I = P • r • t = ____________ • ____________ • ____________ = $ __________
Total Amount = P + I = 4500 + ____________ = ____________
So, after 2.5 years, the total amount in the account was $ _________________.
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384
Holt McDougal Mathematics
Name
Date
LESSON
Class
Student Worksheet
6-7 Simple Interest
Problem 1
Use this diagram to help you to set up an equation to find a percent.
Simple Interest
THE
THEUNITED
UNITEDSTATES
STATESOF
OFAMERICA
AMERICA
THE
THEUNITED
UNITEDSTATES
STATESOF
OFAMERICA
AMERICA
THE
THEUNITED
UNITEDSTATES
STATESOF
OFAMERICA
AMERICA
THE
THEUNITED
UNITEDSTATES
STATESOF
OFAMERICA
AMERICA
P
•
r
•
t
Principal
Rate
Time
Amount of
money
borrowed or
invested
Interest rate
written as a
percent
Number of
years money
in borrowed or
invested
THIS NOTE IS LEGAL TENDER
FOR ALL DEBTS, PUBLIC AND
THISPRIVATE
NOTE IS LEGAL TENDER
FOR ALL DEBTS, PUBLIC AND
THISPRIVATE
NOTE IS LEGAL TENDER
FOR ALL DEBTS, PUBLIC AND
THISPRIVATE
NOTE IS LEGAL TENDER
12
FOR ALL DEBTS, PUBLIC AND PRIVATE
12
12
A
12
12
12
12
12
12
WASHINGTON, D.C.
A
12
WASHINGTON, D.C.
H 293
A
L70744629F
L70744629F
L70744629F
L70744629F
12
L70744629F
L70744629F
L70744629F 12
L70744629F
WASHINGTON, D.C.
WASHINGTON, D.C.
A
H 293
H 293
SERIES
1985
H 293
12
SERIES
1985
ONE DOLLAR
ONE DOLLAR
ONE DOLLAR
ONE DOLLAR
12
SERIES
1985
12
SERIES
1985
12
Interest for
Tristan’s
loan
$14,500
7%
•
Divide rate by 100
to get decimal.
I 14,500 • 0.07 • 5
I 5075
Think and Discuss
5 years
•
So, Tristan will pay $5075 in
simple interest for his loan.
1. In Problem 1, would Tristan pay less simple interest if he paid off
the loan in 4 years? Explain.
2. Would it be in Tristan’s best interest to pay off the loan in
4 years instead of 5 years? Explain.
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88
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-2
Practice A
Integer Exponents
Simplify. Write in decimal form.
1. 101
_______________
5. 100
_______________
9. 107
_______________
2. 106
3. 102
_______________
4. 101
_______________
________________
7. 105
6. 103
_______________
8. 106
_______________
11. 103
10. 104
_______________
________________
12. 105
_______________
________________
Simplify.
13. (2)3
_______________
17. 52
_______________
14. 34
15. (4)2
_______________
16. 24
_______________
18. 63
19. (9)2
_______________
________________
20. (3)3
_______________
21. 8 30 + 21
________________
22. 4 + (6)0 41
_______________
_______________
23. 3(9)0 + 42
24. 6 + (5)2 (4 + 3)0
_______________
_______________
25. One centimeter equals 102 meter. Simplify 102.
________________________________________________________________________________________
26. The area of a square is 104 square feet. Simplify 104.
________________________________________________________________________________________
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189
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-2
Reading Strategies
Using Patterns
The pattern in this table will help you
evaluate powers with exponents.
Column 1
Column 2
Column 3
23 = 8
33 = 27
43 = 64
22 = 4
32 = 9
42 = 16
21 = 2
31 = 3
41 = 4
20 = 1
30 = 1
40 = 1
1
3
1
3 2 =
9
1
4
1
4 2 =
16
Look at the pattern of the products in
the first column. You see that as you
move down the column the products
are getting smaller. That is because
there is one less factor. Each product
is divided by 2 to get the next product.
1
2
1
2 2 =
4
2 1 =
Look at the second and third
columns to answer Exercises 1–6.
3 1 =
4 1 =
1. What is the base in Column 2? ______________________
2. What is the product divided by
each time to get the next product? ______________________
3. What is 1 ÷ 3? ______________________
4. What is the base in Column 3? ______________________
5. What number is the product divided
by each time to get the next product? ______________________
6. What is
1
÷ 4? ______________________
4
Complete the table using the
table above as a guide.
Column 1
Column 2
Column 3
53 = 125
63 = 216
103 = 1000
52 =
62 =
102 = 100
=5
=6
= 10
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195
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-2
Review for Mastery
Integer Exponents
To rewrite a negative exponent,
move the power to the denominator
of a unit fraction.
5 2 = 1
52
Complete to rewrite each power with a positive exponent.
1
1. 73 =
2. 95 =
1
3. 134 =
1
Complete each pattern.
4. 101 =
102 =
1
= 0.1
10
1
102
=
1
= 0.01
100
103 = _______________________________
6. 31 =
1
3
3 2 =
1
3
2
5. 51 =
1
5
5 2 =
1
52
=
1
1
=
5•5
25
53 = _______________________________
7. (4)1 = ______
=
1
1
=
3•3
9
(4)2 = _____________________________
33 = _______________________________
(4)3 = ________________________________
Simplify.
8. 23 =
1
= ____________
9. (6)2 =
1
= ____________
10. 42 =
1
= ____________
11. (3)3 =
1
= ____________
12. 62 = ____________
13. (2)3 = ____________
14. 63 = ____________
15. (5)2 = ____________
16. 24 = ____________
17. (9)1 = ____________
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192
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
4-2 Integer Exponents
Problem 1
3
(–2)
CAUTION
(–2)3
1
1
(–2)3
1
1
(–2) • (–2) • (–2) –8
p!
Hel
When you flip, the sign falls off, the exponent!
Try it on a calculator
( (
or
Problem 2
Which operation is done first?
Think: start inside parentheses
2 (–7)0 (4 2)–2 2 (–7)0 (6)–2
1
Think: (–7)0 1
21
36
1
and 6–2 2
6
Think and Discuss
1. What clue in Problem 1 tells you that you need to find the
reciprocal?
2. If (–7)0 in Problem 2 was (7)0 would the answer be different? Explain.
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44
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-3
Practice A
Scientific Notation
Write each number in standard notation.
1. 1.76 101
2. 8.9 103
_______________
5. 5.8 10
4
_______________
6. 8.1 10
_______________
9. 5.0 10
3
3. 6.2 102
_______________
5
7. 3.8 10
_______________
10. 3.12 10
_______________
4. 1.01 102
________________
4
8. 2.03 103
_______________
5
11. 7.6 10
_______________
________________
2
12. 8.54 105
_______________
________________
Write each number in scientific notation.
13. 376,000
_______________________
16. 1006
_______________________
19. 0.0107
_______________________
22. 250,800
_______________________
14. 9,580,000
15. 650
________________________
17. 29
________________________
18. 0.0061
________________________
20. 0.0002008
________________________
21. 0.00053
________________________
23. 0.000094
________________________
24. 0.00086
________________________
________________________
25. Earth is about 93,000,000 miles from the Sun. Write this number
in scientific notation.
________________________________________________________________________________________
26. The diameter of Earth is about 1.276 104 kilometers.
The diameter of Venus is about 1.21 104 kilometers. Which
planet has the greater diameter, Earth or Venus?
________________________________________________________________________________________
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197
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-3
Reading Strategies
Organization Patterns
You can use powers of 10 to write very large or very small numbers
in a shortened form. This efficient method is called scientific
notation. It is also useful in performing multiplication and division of
very large and very small numbers.
Standard form
348,000,000
Scientific notation
=
3.48 108
8 places left
Move the decimal point to
create a number between
1 and 10.
The number of places the
decimal point is moved to the
left is the positive exponent.
Standard form
0.00035
Scientific notation
=
3.5 104
4 places right
Move the decimal point to
create a number between
1 and 10.
The number of places the
decimal point is moved to the
right is the negative exponent.
Use 0.000078 to answer Exercises 1–4.
1. How many places must you move the decimal
point to create a number between 1 and 10? _________________________________________
2. Which direction will you move the decimal point? _____________________________________
3. Will the exponent be negative or positive? ____________________________________________
4. Write the number in scientific notation. _______________________________________________
Use 312,000,000 to answer Exercises 5–7.
5. How many places must you move the decimal
point to create a number between 1 and 10? _________________________________________
6. Which direction will you move the decimal point? _____________________________________
7. Will the exponent be negative or positive? ____________________________________________
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203
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-3
Review for Mastery
Scientific Notation
Standard Notation
430,000
0.0000057
(
Scientific Notation
1st factor is
between 1 and 10.
2nd factor is an
integer power of 10.
4.3 105
5.7 106
positive integer for large number
negative integer for small number
)(
)
To convert from scientific notation, look at the power of 10 to tell
how many places and which way to move the decimal point.
Complete to write each in standard notation.
1. 4.12 106
2. 3.4 105
Is the exponent positive or negative?
_______________
_______________
Move the decimal point right or left?
How many places?
_______________
_______________
Write the number in standard notation.
_______________
_______________
Write each number in standard notation.
3. 8 105
____________________________________
4. 7.1 104
5. 3.14 108
____________________________________
___________________________________
To convert to scientific notation, determine the factor between 1 and
10. Then determine the power of 10 by counting from the decimal
point in the first factor to the decimal point in the given number.
Complete to write each in scientific notation.
6. 32,000,000
7. 0.0000000712
What is the first factor?
_______________
________________
From its location in the first factor, which
way must the decimal move to its location
in the given number? How many places?
_______________
________________
Write the number in scientific notation.
_______________
________________
Write each number in scientific notation.
8. 41,000,000
____________________________________
9. 0.0000000643
____________________________________
10. 1,370,000,000
___________________________________
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200
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
4-3 Scientific Notation
Problem 1
Think about the number line.
3.12 109
Which direction should you
move the decimal point?
2
1
0
To the left
1
2
To the right
9 is “” so move the decimal point to the right.
Problem 2
Write 0.0000003 in scientific notation.
0.0000003.
Decimal point moves 7 places.
Is the absolute value of the number 1 or 1?
If 1, then 3 107
If 1, then 3 107
|0.0000003| 1
So 0.0000003 3 107.
Think and Discuss
1. If a number is extremely large will the exponent be positive or
negative if you write the number in scientific notation?
2. Write 3 107 in decimal form.
3. Is 3 107 grams more likely to be the weight of a car or the
weight of an eyelash? Explain.
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46
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class__________________
LESSON
4-4
Practice A
Laws of Exponents
Multiply. Write the product as one power.
1. 22 • 23
2. 35 • 32
_______________
1
5. 8 • 8
1
3. 13 • 15
________________
4
6. 7 • 7
_______________
4. 54 • 53
________________
5
1
7. 12 • 12
________________
2
________________
8. n3 • n8
________________
________________
Divide. Write the quotient as one power.
9.
25
22
10.
_______________
13.
58
56
_______________
10 4
11.
103
________________
14.
46
12.
43
________________
249
15.
243
________________
(6)8
________________
(3)4
________________
16.
(6)5
(3)6
b7
b5
________________
Simplify.
17. (32)4
_______________
2 3
21. (5 )
_______________
18. (63)1
19. (45)0
________________
20. (82)3
________________
0 4
4 2
22. (7 )
24. (s5)2
23. (9 )
________________
________________
________________
________________
3
25. The mass of a male African elephant is about 7 10 kg. What is the combined
mass of a herd of 80 male African elephants? Write your answer in scientific
notation.
________________________________________________________________________________________
26. The Haywood Paper Company has 52 warehouses. Each
warehouse holds 55 boxes of paper. How many boxes of paper
are stored in all the warehouses? Write the answer as one
power.
________________________________________________________________________________________
27. Write the expression for 5 used as a factor eight times being
divided by 5 used as a factor six times. Simplify the expression
as one power.
________________________________________________________________________________________
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205
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Reading Strategies
LESSON
4-4
Organization Patterns
There are some rules that make multiplying or dividing exponents
with the same base easier.
To multiply powers with the same base, add exponents.
(4 • 4)
•
(4 • 4 • 4)
=
4•4•4•4•4
42
•
43
=
45
The base of 4 is the same, so: 42 • 43 = 42 + 3 = 45.
To divide powers with the same base, subtract the exponents.
6•6•6•6•6•6
6•6•6
=
6•6•6•6•6•6
6 •6•6
=
66
63
63
The base of 6 is the same, so:
66
6
3
= 66 3 = 63
Answer each question.
1. What is the base for 32?
______________________________________________________________
2. What is the base for 34 ?
______________________________________________________________
3. Are the bases the same for these powers?
4. Write all the factors for 32 • 34.
___________________________________________
________________________________________________________
5. Add the exponents for 32 and 34 and
rewrite the number using the same base.
____________________________________________
6. Are the answers for Exercise 4 and Exercise 5 the same?
7. Are the bases the same for 55 ÷ 52?
___________________________
__________________________________________________
8. Subtract the exponents and rewrite the problem. _____________________________________
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211
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-4
Review for Mastery
Laws of Exponents
To multiply powers
with the same base,
keep the base and add
exponents.
To divide powers with
the same base,
keep the base and
subtract exponents.
To raise a power
to a power,
keep the base and
multiply exponents.
xa • xb = xa+b
xa ÷ xb = xab
(xa)b = xab
45 • 42 = 45 + 2 = 47
45 ÷ 42 = 45 2 = 43
(45)2 = 45(2) = 410
83 • 8 = 83 + 1 = 84
83 ÷ 8 = 831 = 82
Complete to see why the rules for exponents work.
1. 45 • 42 = ( _____ ) ( _____ ) ( _____ ) ( _____ )( _____ ) • ( _____ )( _____ ) = 4_____
2. 83 • 8 = ( _____ ) ( _____ ) (_____ ) • (_____ ) = 8_____
3. 45 ÷ 42 =
4. 83 ÷ 8 =
45
4
2
4 •4•4•4•4
= 4_____
4 •4
=
8 •8•8
83
=
= 8_____
8
8
+2+ 2
5. (42)3 = 42 • 42 • 42 = 42
= 42(3) = 4________
Complete to write each product or quotient as one power.
+
6. 123 • 122 = 123 2 = 12_____
8.
76
7
2
7. 94 • 93 = 9________ = 9_____
= 76–2 = 7_____
9.
126
12
4
= 12________ = 12_____
Write each product or quotient as one power.
10. 104 • 106 = _________
13.
15 6
15 2
= _________
11. 55 • 5 = _________
14.
95
= _________
9
12. 45 • 4 • 43 = _________
15.
210
22
= _________
Simplify.
16. (53)4 = 53(4) = _________
17. (62)4 = 62(4) = _________
18. (25)2 = _________
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208
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
4-4 Laws of Exponents
Problem 1
Can you see a relationship?
72 • 72 74
49 • 49 74
2401 74
2401 2401
Problem 2
Can you see a relationship?
(75)3 715
(75) • (75) • (75) 715
(75 5 5) 715
715 715
to
Yes!
power
a
e
is
To ra
ly
r, multip
a powe ents!
on
the exp
Think and Discuss
1. Does x m • y n xy m n? Explain.
2. Sylvia says that (154)2 simplifies to 156. Is she correct? Explain.
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48
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-5
Practice A
Squares and Square Roots
Find the two square roots of each number.
1. 16
2. 49
_______________
3. 1
_______________
5. 100
_______________
6. 4
_______________
4. 25
7. 81
_______________
________________
8. 64
_______________
________________
Simplify each expression.
9.
8 +1
_______________
13.
36 + 10
64
16
11.
_______________
15.
_______________
12.
49 4
19.
_______________
100
4
_______________
31 + 5
________________
16.
_______________
18. 5 9
_______________
18 2
_______________
14. 15 25
_______________
17.
76
10.
16 + 9
________________
20.
3 81
________________
Switzerland’s flag is a square, unlike
other flags that are rectangular.
21. If the flag of Switzerland has an area
of 81 ft2, what is the length of each of
its sides? (Hint: s =
A)
_______________________________________
22. If the lengths of the sides of a
Switzerland flag are 10 ft, what is the
area of the flag? (Hint: A = s 2)
_______________________________________
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213
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Reading Strategies
LESSON
4-5
Connect Words with Symbols
A square root produces a given number when multiplied by itself.
The large square shown below is 4 squares long on each side and
has 16 squares. 4 times 4 equals 16. 4 is the square root of 16.
The 4 4 square can be described with symbols and with words.
Symbols
Symbols
Words
4 • 4 = 16
42 = 16
Four squared equals sixteen.
This sign represents square root:
16 = 4
Read “The square root of 16 equals 4.”
25 = 5
Read “The square root of 25 equals 5.”
Compare the symbols for “squared” and “square root.”
42 = 16 and
2
5 = 25 and
16 = 4
25 = 5
Write in words.
1. 62
2.
____________________________________________
36
_______________________________________________
Answer each question.
3. What is the square root of 36?
4. What is the square root of 100?
5. What is 72?
_______________________________
________________________________
_____________________________
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219
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-5
Review for Mastery
Squares and Square Roots
A perfect square has two identical factors.
25 = 5 5 = 52 or 25 = (5) (5) = (5)2
then 25 is a perfect square.
Tell if the number is a perfect square.
If yes, write its identical factors.
1. 121 ___________________________
2. 200 ___________________________
3. 400 ___________________________
Since 52 = 25 and also (5)2 = 25,
both 5 and 5 are square roots of 25.
The principal square root of 25 is 5:
25 = 5 and 25 = 5
25 = 5
Write the two square roots of each number.
4.
81 = _____________
625 = _____________
5.
81 = __________
6.
625 = __________
169 = ______________
169 = __________
Write the principal square root of each number.
7.
144 = ___________
8.
6400 = ___________
9.
10,000 = ____________
5 100 3
5(10) 3
50 3
47
Use the principal square root when
evaluating an expression. For the
order of operations, do square root
first, as you would an exponent.
Complete to simplify each expression.
10. 3 144 20
3 ________ 20
________
20
________
11.
25 + 144 + 13
__________
__________
+ 13
+ 13
________
12.
1
100
+
25
2
100
25
+
1
2
1
5
2
1
__________ +
2
+
_________
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216
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
4-5 Squares and Square Roots
Problem 1
Think: What number
times itself equals 81?
81
81
9
81
9
4 4 16
5 5 25
6 6 36
7 7 49
8 8 64
9 9 81
4 4 16
5 5 25
6 6 36
7 7 49
8 8 64
9 9 81
Problem 2
Order of Operations:
1. Parentheses
2. Exponents and roots
3. Multiply and divide from left to right.
4. Add and subtract from left to right.
325
4
3•5 4
15
4
19
Square root first.
Multiply.
Add.
Think and Discuss
1. Why is 52 read as “five squared”?
2. You know that the product of two positive numbers is positive
() and the product of two negative numbers is positive
(3 • 3 9). Use these rules to explain why 9
is
undefined.
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50
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class__________________
LESSON
4-5
Problem Solving
Squares and Square Roots
Write the correct answer.
1. For college wrestling competitions,
the wrestling mat must be a square
with an area of 1,764 square feet.
What is the length of each side of the
wrestling mat?
2. For high school wrestling
competitions, the wrestling mat must
be a square with an area of 1,444
square feet. What is the length of
each side of the wrestling mat?
(Hint: A = s2)
Solution:
1,444 = ___________
1,764 = 42 feet
___________ feet
3. Elena has a large sheet of square
paper that is 169 square inches. How
many squares can she cut out of the
paper that are 4 inches on each side?
4. James has a square area rug that is
132 square feet. In his new house,
there are three rooms. Room one is
11 feet by 11 feet. Room two is 10
feet by 12 feet and room three is 13
feet by 13 feet. In which room will the
rug fit?
________________________________________
________________________________________
Choose the letter for the best answer.
6. To create a square patchwork quilt,
square pieces of material are sewn
together to form a larger square.
Which number of smaller squares
can be used to create a square
patchwork quilt?
5. A square picture frame measures
36 inches on each side. The actual
wood trim is 2 inches wide. The
photograph in the frame is surrounded
by a bronze mat that measures
5 inches. What is the maximum area
of the photograph?
A 35 squares
A 841 sq. inches
C 64 squares
B 84 squares
B 961 sq. inches
C 484 sq. inches
8. A box of tile contains 12 square tiles.
If you tile the largest possible square
area using whole tiles, how many tiles
will you have left from the box?
7. A can of paint claims that one can will
cover 400 square feet. If you painted a
square with the can of paint, how long
would it be on each side?
A 200 feet
C 20 feet
A 9
C 6
B 3
B 25 feet
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62
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-6
Practice A
Estimating Square Roots
Each square root is between two consecutive integers. Name
the integers. Explain your answer.
1.
2.
10
_______________________________________
3.
________________________________________
4.
19
_______________________________________
5.
8
33
________________________________________
6.
15
_______________________________________
39
________________________________________
Approximate each square root to the nearest hundredth.
7.
32
8.
_______________
11.
22
9.
59
_______________
12.
_______________
10.
118
_______________
13.
155
_______________
230
________________
14.
43
_______________
181
________________
Use a calculator to find each value. Round to the nearest tenth.
15.
12
16.
_______________
19.
38
_______________
20.
_______________
23.
54
_______________
17.
18
18.
_______________
21.
45
_______________
24.
7
8
________________
22.
_______________
25.
27
_______________
40
_______________
24
22
________________
26.
48
________________
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221
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class__________________
LESSON
4-6
Reading Strategies
Follow a Procedure
The numbers 16 and 25 are called perfect squares. Each has an
integer as its square root. To find the square root of a perfect
square, ask yourself what number multiplied by itself equals the
perfect square.
Some Perfect Squares
1
4
9
16
36
49
64
81
100
121
144
169
25
1. What number times itself equals 16? ___________________________
2. What is the square root of 16? ___________________________
3. What number times itself equals 25? ___________________________
4. What is the square root of 25? ___________________________
Use these steps to estimate the square root of a number that is not a
perfect square.
What is
45 ?
Step 1
Identify a perfect square that is a little more than 45.
The square root of 49 = 7.
Step 2
Identify a perfect square that is a little less than 45.
The square root of 36 = 6.
49
36
Step 3
The estimate of
45 is between 6 and 7.
Use the steps above to help you estimate the square root of 90.
5. Which perfect square is a little more than 90? _________________
6. What is the square root of 100? _________________
7. Which perfect square is a little less than 90? _________________
8. What is the square root of 81? _________________
9. What is your estimate of the square root of 90?
________________________________________________________________________________________
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227
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Review for Mastery
LESSON
4-6
Estimating Square Roots
To locate a square root between two consecutive integers, refer to the table.
Number
Square
1
1
2
4
3
9
4
16
5
25
6
36
7
49
8
64
9
81
10
100
Number
Square
11
121
12
144
13
169
14
196
15
225
16
256
17
289
18
324
19
361
20
400
256
< 260
Locate 260 between two integers.
260 is between the perfect squares 256 and 289:
256 <
So:
And:
16
< 289
260 <
<
289
260 < 17
Use the table to complete the statements.
1.
______
< 39
______
<
______
<
< ______
______
< 130 < ______
39 < ______
______
< 130 < ______
39 < ______
______
< 130 < ______
2.
After locating a square root between two consecutive integers, you can
determine which of the two integers the square root is closer to.
27 is between the perfect squares 25 and 36:
25 < 27
< 36
25 <
So:
And:
The difference between 27 and 25 is 2;
the difference between 36 and 27 is 9.
So,
<
5
27 <
36
27 < 6
25 < 27 < 36
27, is closer to 5.
2
9
Complete the statements.
3. 100
< 106
< 121
4.
_______ <
250
< _______
_______
< 106 < _______
_______
<
250 < _______
_______
< 106 < _______
_______
<
250 < _______
250 _______ = _______
106 100 = _______
_______ 121 106 = _______
106 is closer to ______ than ______
250 = _______
250 is closer to _______ than _______
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224
Holt McDougal Mathematics
Name
Date
LESSON
Class
Student Worksheet
4-6 Estimating Square Roots
Problem 1
52 25 and
62 36
Is 5 30 6?
30 5.47722
Use a calculator. 5 5.4772255… 6
Yes, 30 is
between
5 and 6.
Problem 2
冑 700
⬇
26.5
It reads "the
square root of
700 is about
26.5."
What
is this?
⬇
means ABOUT
Think and Discuss
1. The square root of 5 is between 2 and 3. Which is a more
precise statement 2 5
3 or 5
2.2? Why?
2. Why do you use the term “about” when reading the answer to
Problem 2?
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52
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class__________________
LESSON
4-7
Practice A
Operations with Square Roots
Simplify.
1. 8 2 + 3 2
2. 10 5 6 5
________________________
4.
7.
10.
3. 6 7 + 7 + 2 7
________________________
________________________
8 2
5. 3 10 10
6. 5 3 27
____
3 ______
___ ___
_____
3 ______
___ ___
________
_______
8.
45
9.
32
___ 5
___ 2
____ 5
____ 2
________
_______
27 + 48
11.
___ 3 + ___ 3
50 18
________
300
______
27 + 6 3
12.
________
___ 3 + ___ 3
___ 3
13. The length of a room is exactly
242 feet. Simplify the length.
________________________________________________________________________________________
14. Pipe A has a width of 125 centimeters, and Pipe B has a width
of 80 centimeters. Write the difference in the two widths in
simplified form.
________________________________________________________________________________________
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229
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-7
Reading Strategies
Follow a Procedure
Use the following procedure to simplify square roots.
Step 1: List all the factors of the number.
Step 2: Ask yourself: Are any of the factors perfect squares?
If yes, circle the greatest perfect square factor.
If no, stop, the expression is already simplified.
Step 3: Write the number under the square root symbol as a product of two numbers,
where one of the numbers is the number you circled in Step 2.
Step 4: Use the Multiplication Property of Square Roots to write the expression as two
separate square roots.
Step 5: Take the square root of the perfect square.
1. Follow the procedure to simplify
24 .
Steps 1 and 2: ________________________________________
Step 3:
____ ____
Step 4:
____ ____
Step 5: ____ ____
2. Follow the procedure to simplify
54 .
Steps 1 and 2: ________________________________________
Step 3: ______________________
Step 4: ______________________
Step 5: ______________________
Simplify.
3.
12
_______________
4.
5.
27
_______________
21
_______________
6.
63
_________________
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235
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Review for Mastery
LESSON
4-7
Operations with Square Roots
When the numbers under the square root symbols are the same, you can add or
subtract them.
4 7 +9 7
Add the numbers
outside the square
root symbol.
13 7
Simplify.
1. 3 2 + 8 2
2. 9 6 2 6
___________________
3.
_______________
5 +7 5
_______________
4. 13 10 4 10
_________________
The numbers under the square root symbols do not have to be the same in order to
multiply them.
5 20
3 7
Simplify when
needed.
100
21
10
Simplify.
5.
2 2
6. 4 5 5
_______________
7.
_______________
8 3 2
_______________
8. 8 3 3
_________________
To simplify square roots, write the number under the square root symbol as a product.
Make one of the factors a perfect square.
90
9 is a perfect square. The
square root of 9 is 3.
Leave the factor that is not
a perfect square under the
square root symbol.
9 10
3 10
Simplify.
9.
45
_______________
10.
11.
12
_______________
600
_______________
12. 3 50
_________________
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232
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
4-7 Operations with Square Roots
Problem 1
Can I add radicands?
No!
I’ll add like apples.
2 10 5 10 ?
10 is the radicand.
2 10
7 square roots of 10
5 10
( 10 10 ) ( 10 10 10 10 10 )
So, 2 10 5 10 7 10
Easier: 2 10 5 10 (2 5) 10
Use the Distributive Property.
7 10
Problem 2
Can I multiply radicands?
Yes!
12 •
3 ?
12 •
12 and 3 are the
radicands.
3 12 • 3
36
6
6 36 because 6 • 6 36.
Think and Discuss
1. Tami said that 9
16
Was she right? Explain.
25 because 9 16 25.
2. Show how you can simplify the expression
11
7 35
57 45 .
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54
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-8
Practice A
The Real Numbers
Write all names that apply to each number.
1. 3.2
2.
2
5
3. 12
_______________________
________________________
________________________
_______________________
________________________
________________________
4
2
4.
5. 20
6.
16
_______________________
________________________
________________________
_______________________
________________________
________________________
State if the number is rational, irrational, or not a real number.
7. 0
8.
_______________
11.
3
4
_______________
12.
_______________
4
9.
7
10.
_______________
________________
13. 49
25
_______________
9
0
14.
_______________
11
________________
Find a real number between each pair of numbers.
15. 3
1
2
and 3
3
3
_______________________
16. 2.16 and
11
5
17.
________________________
1
1
and
8
5
________________________
18. Give an example of an irrational number that is greater than 0.
________________________________________________________________________________________
19. Give an example of a number that is not real.
________________________________________________________________________________________
20. Give an example of a rational number between
1
and
2
1.
________________________________________________________________________________________
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237
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-8
Reading Strategies
Use a Venn Diagram
You know that rational numbers can be written in fraction form as
an
integer
. Rational numbers include:
integer
• Decimals
• Fractions
• Integers
• Whole Numbers
This diagram of rational numbers expressed in different forms helps
you see how they are related.
From this picture you can say:
1. 0.4 is a rational number, but it is not an integer or
___________________________
2.
100 = 10. It is a rational number, it is ___________________________,
and it is a whole number.
3. 3 is a rational number and an integer, but it is not
___________________________.
4. 2.6 is a rational number, but it is not ___________________________ or a
whole number.
Numbers that are not rational are called irrational numbers. For
example,
3 is an irrational number. It is a decimal that does not
terminate or repeat.
3 = 1.7320508…
Write all names that apply to each number: rational, irrational,
integer, or whole number.
5. 2.236068…
_____________________________________________.
6. 7
_______________________________________________________.
7. 328
_______________________________________________________.
8. 2
2
3
_______________________________________________________.
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243
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-8
Review for Mastery
The Real Numbers
The set of rational numbers contains
all integers, all fractions, and decimals
that end or repeat.
Irrational numbers can only be written
as decimals that do not end or repeat.
Real Numbers
Rational Numbers
Irrational Numbers
Together, the rational numbers and the
irrational numbers form the set of real numbers.
Square roots of numbers that are Square roots of numbers that are
perfect squares are rational.
not perfect squares are irrational.
25 = 5
3 = 1.732050807. . .
Tell if each number is rational or irrational.
1.
7
_______________
2.
3.
81
_______________
4.
169
_______________
2
101
________________
2
The square of a nonzero number is positive. 3 = 9 and (3) = 9
So, the square root of a negative number is not a real number.
9 is not a real number.
Tell if each number is real or not real.
5. 8
_______________
6. 8
7.
_______________
8
25
8.
_______________
________________
Between any two real numbers, there is always another real number.
One way to find a number between is to find the number halfway
between.
1
2
To find a real number between 7 and 7 ,
5
5
3
1
2
3
divide their sum by 2: 7 + 7 = 14 ÷ 2 = 7
10
5
5 5
Find a real number between each pair.
9. 8
4
3
and 8
7
7
_______________
10. 1.6 and 1.7
11. 3
_______________
7
2
and 3
9
9
_______________
12. 6
1
3
and 6
2
4
________________
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240
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
4-8 The Real Numbers
Problem 1
Before you classify the numbers, look closely at each number.
15
3
–
0
1
–
9
The square root of a negative
number is undefined.
Divison by zero is undefined.
CAUTION
13
Always look for undefined
expressions. They look obvious!
Problem 2
1 1–
2 )2
(1 –
3
3
Hmm, Where have
I seen this concept before?
It is the average, or
mean.
Think and Discuss
1. What is the only set of numbers that irrational numbers can belong to?
2. How many sets of numbers does a whole number belong to?
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56
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
4-8
Problem Solving
The Real Numbers
Write the correct answer.
1. Twin primes are prime numbers that
differ by 2. Find an irrational number
between twin primes 5 and 7.
2. Rounded to the nearest
ten-thousandth, = 3.1416 . Find a
rational number between 3 and .
Solution:
Possible answer: _________
You need to find an irrational number
between 5 and 7.
Since 52 = 25, 72 = 49, and 62 = 36,
try the square root of a number
between 25 and 49 that is not 36.
Possible answer:
31
3. One famous irrational number is e.
Rounded to the nearest
ten-thousandth e 2.7823 . Find a
rational number that is between 2
and e.
4. Perfect numbers are those for which
the divisors of the number sum to the
number itself. The number 6 is a
perfect number because
1 + 2 + 3 = 6. The number 28 is
also a perfect number. Find an
irrational number between 6 and 28.
________________________________________
_______________________________________
Choose the letter for the best answer.
6. Which is an irrational number?
5. Which is an integer?
A the number half-way between 6
and 7
A a number that can be expressed
as a fraction
B the money in an account if the
balance was $213.00 and $21.87
was deposited
B the length of a side of a square
with area 2 cm2
C the square root of a negative
number
C the net yardage after plays that
resulted in a 15 yard loss, 10 yard
gain, 6 yard gain and 5 yard loss
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68
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-3
Practice A
Adding and Subtracting Rational Numbers
Name a common denominator for each sum or difference.
Do not solve.
1.
1
3
+
2
4
2.
_______________
5. A statue 8
that is 1
1
4
+
3
9
3.
_______________
2
3
3
8
4.
_______________
1
1
2
6
________________
6. During the 19th Olympic Winter
Games in 2002, the United States
4-man bobsled teams won silver and
bronze medals. USA-1 sled had a
total time of 3 min 7.81 sec. The
USA-2 sled had a total time of 3 min
7.86 sec. What is the difference in the
time of the two runs?
5
in. high rests on a stand
16
3
in. high. What is the total
16
height?
_______________________________________
________________________________________
Add or subtract. Write each answer in simplest form.
7.
2
4
+
9
9
_______________
11.
2
1
5
2
_______________
8.
5
3
+
12
12
9.
_______________
12. 2
9
7
10
10
10.
_______________
1
1
+1
3
2
13. 3
_______________
1 5
+ 1
4 6 _______________
8
11
15
15
________________
14.
3
11
4 12
________________
15. Mr. Martanarie bought a new lamp and lamppost for his home.
5
1
The pole was 6 ft tall and the lamp was 1 ft in height. How
8
4
tall were the lamp and post together?
________________________________________________________________________________________
Simplify each expression.
16. 1
1 1
+
2 2
_______________
17. 1
1
1
+ 2
2
18. 5
_______________
1
1
2
6
3
_______________
19. 12
3
3
5
4
12
________________
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98
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-3
Reading Strategies
Use a Graphic Aid
It is easy to add and subtract fractions with common denominators.
3 eighths + 4 eighths = 7 eighths
3
8
+
4
8
=
8 ninths 3 ninths = 5 ninths
8
9
7
8
3
9
=
5
9
Adding fractions with unlike denominators requires more steps. The
picture below will help you understand adding fractions with unlike
1 1
denominators. + = ?
2 4
In order to add
1 1
+ , you must find a common denominator.
2 4
1. What are the denominators in this problem?
_________________
2. To find a common denominator, one-half can be changed
into fourths. How many fourths are there in one-half?
_________________
1
to fourths.
2
4. You can now add, because you have a common denominator.
3. Change
_________________
_________________
To subtract fractions with unlike denominators, you must find a
common denominator. The picture below will help you understand
5 1
finding a common denominator. = ?
6 3
5. What are the denominators in this problem?
_________________
To find a common denominator, you will change to sixths.
6. How many sixths are in one-third? Write the fraction.
_________________
7. You can now subtract the fractions.
_________________
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105
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-3
Review for Mastery
Adding and Subtracting Rational Numbers
To add fractions that have the same denominator:
• Use the common denominator for the sum.
• Add the numerators to get the numerator of the sum.
• Write the sum in simplest form.
3 1+ 3 4
1
1
+
=
=
=
8
8
8
8
2
To subtract fractions that have the same denominator:
• Use the common denominator for the difference.
• Subtract the numerators.
Subtraction is addition of an opposite.
• Write the difference in simplest form.
3 1 3 +1 4
2
=
=
=
6 6
6
6
3
Complete to add the fractions.
1.
3
4
+
= _____ = _____
14
14
3. 2.
4
2
+ = _____ = _____
10 10 5.
3
9
= _____ = _____
15 15 3
5
+ = _____ = _____
12 12 Complete to subtract the fractions.
4.
8 2
= _____ = _____
9 9
10 2 = _____ = _____
24 24 To add or subtract decimals, line up the decimal points and then add or subtract from
right to left as usual.
6. 12.83
+ 24.17
35.78
14.55
37.00
21.23
Complete to add the decimals.
7. 14.23 + 3.56 = _________________
8. 44.02 + 8.07 = _________________
9. 1.39 + 13.6 = _________________
Complete to subtract the decimals.
10. 124.33 13.16 = _________________
11. 33.47 0.6 = _________________
12. 25.15 25.06 = _________________
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101
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-3
Review for Mastery
Adding and Subtracting Rational Numbers (continued)
To add fractions with different denominators, first write the fractions
with common denominators. To find the LCD of denominators 5 and
6, list the multiples of each.
Multiples of 5: 5, 10, 15, 20, 25, 30
Multiples of 6: 12, 18, 24, 30
So, the LCD of 5 and 6 is 30.
Complete to find the LCD for each set of denominators.
13. The LCD of 6 and 4 is: ___________________________
Multiples of 6: ___________________________
Multiples of 4: ___________________________
14. The LCD of 3 and 7 is: ___________________________
Multiples of 3: ___________________________
Multiples of 7: ___________________________
To add fractions with different denominators:
Add:
1 1 1• 3 3
+ =
=
2 3 2•3 6
1• 2 2
=
3•2 6
5
=
6
Complete to add fractions. Simplify.
15.
+
1
=
4 20
16.
3
=
4 16
17. 5
1
=5
3
24
3
=
5 20
+
5
=
16 16
+2
5
=2
8
24
= ________
= ________ = ________
= ________
Add or subtract fractions. Simplify.
18.
1 7
+
=
4 20
19.
4 1
=
9 5
20.
8 1
=
15 4
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102
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
2-3 Adding and Subtracting Rational Numbers
Problem 1
7
13
11
13
Add .
There are a total of 18
shaded squares.
Problem 2
Write a mixed number as an improper fraction.
1
7 1(8) 7
8
8
Multiply the whole
number and the denominator.
Add the numerator.
Keep the
denominator.
Mixed Number
7
15
1 8
8
Improper Fraction
Think and Discuss
1. Explain how to add rational numbers that have the same
denominator.
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24
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-4
Practice A
Multiplying Rational Numbers
Multiply. Write each answer in simplest form.
1
1. 5 3
2
2. 2 5
_______________
5 2
5. 7 5
_______________
6.
_______________
9.
1 10 2 7 10.
3 1
4 3
1 1
7. 4 3
_______________
________________
1 2
8. 6 3
_______________
3 5
10 18 11.
_______________
14.
2
4. 3 9
_______________
_______________
_______________
1
13. 4 1 2
1
3. 4 6
4 12 5 16 ________________
12.
_______________
3 5
4 8
2 1
15. 3 5 4
_______________
4 24 3 16 ________________
5 3 16. 6 10 _______________
________________
Multiply.
17.
3.2
5
18.
_______________
21.
3.14
0.007
_______________
0.34
0.06
19.
_______________
22.
8.12
9
20.
_______________
6.7
0.8
23.
_______________
0.25
2.4
_______________
4.24
3.5
________________
24.
7.9
2
________________
1
hours for the Lenox family. She was paid $5 an hour. How much
2
did she receive for this babysitting job?
25. Jade babysat 4
________________________________________________________________________________________
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107
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class__________________
LESSON
2-4
Reading Strategies
Use a Visual Model
This rectangle will help you understand how to find the
1 1
1
product of
• . First,
of the rectangle was shaded. Then,
2 3
2
1
the rectangle was divided horizontally into thirds. Then,
was
3
1 1
shaded. The overlap of the shading shows the product of
• .
2 3
1. Into how many parts is the rectangle divided? What fractional
part of the rectangle is each of these parts?
_____________
2. What fractional part of the rectangle has shading that overlaps?
_____________
3. Multiply the numerators and the denominators of the given
fractions.
_____________
Use the rectangle to draw a model for the problem
=
1 1
• .
4 2
4. Draw lines from top to bottom to divide the rectangle into
fourths. Shade one-fourth of the rectangle.
5. Draw a line across the rectangle to divide it into halves.
Into how many parts is the rectangle now divided?
_____________
6. Shade one of the halves.
7. What fractional part of the rectangle was shaded twice?
_____________
8. Multiply the numerators and denominators.
_____________
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113
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-4
Review for Mastery
Multiplying Rational Numbers
3
1
.
4
3
Divide a square into
4 equal parts. Lightly
shade 3 of the 4.
To model
Darken 1 of the
3 shaded parts.
Compare the
1 darkened part to
the original 4.
1
3
1
=
3
4
4
Model each multiplication. Write the result.
1.
2.
1
2
= __________
2
4
3.
3
4
= __________
4
6
2
3
= __________
3
9
To multiply fractions:
1
• Cancel common factors, one in a numerator
and the other in a denominator.
• Multiply the remaining factors in the
numerator and in the denominator.
• If the signs of the factors are the same, the product is positive.
If the signs of the factors are different, the product is negative.
2
3
8
1 2
2
=
=
1 3
3
4
9
1
3
Multiply. Answer in simplest form.
1
4
= _______
2
9
2 9
= _______
7.
3 10 4.
6
2
= _______
7
3
2 27
8. = _______
9 40
5.
3 15
5 17
4
9. 7
6.
= _______
21
= _______
8
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110
Holt McDougal Mathematics
Name
Date
Class
Student Worksheet
LESSON
2-4 Multiplying Rational Numbers
Problem 1
Multiply.
5 12
5
12
5 12
5
12
1 1
1 1
Cancel the common factors.
1
1
1
Multiply.
Problem 2
0.07(4.6) 0.322
Why 3 decimal places?
0.07 (4.6 ) 0.322
21 3
Add the decimal places in the factors.
Think and Discuss
1. Ming multiplies two fractions. The product simplifies to 1. What
do you know about the two fractions?
2. If you multiply 462.0125 and 50.375, how many decimal places
will the product have? How do you know?
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26
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-5
Practice A
Dividing Rational Numbers
Divide. Write each answer in simplest form.
1.
1 3
÷
8
4
2. _______________
5.
1
1
÷
9
3
2
1
÷1
5
2
3.
_______________
6.
_______________
9. 1
5
2
÷
9
3
7. _______________
_______________
4.
_______________
2
4
÷
5
7
10. 1
1
÷
6
3
________________
3
6
÷
5
7
8. _______________
3
÷9
4
11. 2
_______________
3 1
÷ 4 8 1
1
÷
3
4
3 5
÷ 8 6 ________________
12. _______________
5
÷5
8
________________
Find each quotient.
13. 1.53 ÷ 0.3
14. 5.14 ÷ 0.2
_______________
17. 6.54 ÷ 0.03
15. 10.05 ÷ 0.05
_______________
_______________
18. 29.45 ÷ 0.005
_______________
16. 5.28 ÷ 0.4
19. 8.58 ÷ 0.06
_______________
________________
20. 1.61 ÷ 0.7
_______________
________________
Evaluate each expression for the given value of the variable.
21.
10
for x = 0.05
x
_______________________
22.
9.12
for x = 0.2
x
23.
________________________
24. Mr. Chen has a 76-in. space to stack books. Each book is 6
42.42
for x = 1.4
x
________________________
1
in.
3
tall. How many books can he stack in the space?
________________________________________________________________________________________
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115
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-5
Reading Strategies
Focus on Vocabulary
The word reciprocal means an exchange. When two friends
exchange gifts, you might think of the gifts as “switching places.”
In the reciprocal of a fraction, the numerator and denominator
exchange places.
Fraction Reciprocal
2
3
3
2
4
5
5
4
8
1
1
8
1. What does the word reciprocal mean? _________________
2. What is the reciprocal of
7
? _________________
8
3. What is the reciprocal of
6
? _________________
5
The product of a fraction and its reciprocal is always 1.
Fraction • Reciprocal = Product
2 3
6
•
=
=1
3 2
6
4 5
20
•
=
=1
5 4
20
1 8
8
•
=
=1
8 1
8
4. What is the product of
1 7
• ? _________________
7 1
5. What is the product of
2
and its reciprocal? _________________
6
6. What is the reciprocal of
7. What is the product of
1
? _________________
2
1
2? _________________
2
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121
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-5
Review for Mastery
Dividing Rational Numbers
To write the reciprocal of a fraction,
interchange the numerator and denominator.
The product of a number and its reciprocal is 1.
2
3
Fraction
3
2
Reciprocal
2
3
=1
3
2
Write the reciprocal of each rational number.
1. The reciprocal of
3
is:
5
2. The reciprocal of 6 is:
_______________________
________________________
3. The reciprocal of 2
1
is:
3
________________________
To divide by a fraction, multiply by its reciprocal.
2
÷6
3
3
9
÷
5 10
2
1
3
6
3 10
5
9
1
2/ 1
=
3 6/ 3 9
3 10
2
=
3
5 9
1
1
2
1
3
Complete to divide and simplify.
4
4
÷ 16 =
_________ = __________
3
3
5
20
5
3 9
3
6.
÷
=
_______ = __________
7. ÷ = _______ = _______
7
21 7
4 8
4
6.2
Change a decimal divisor to a whole number.
0.7 4.34
0.7. 4.3.4
7 43.4
Using the number of places in the divisor,
4.
3
3
÷ 12 =
_______ = ___________
8
8
5.
move the decimal point to the right in both
the divisor and the dividend.
Rewrite each division with a whole-number divisor.
Then, do the division.
8. 0.6 1.14
10. 0.02 7.12
__________ = _______
9. 0.3 4.56
__________ = _______
11. 0.08 57.28
__________ = _______
__________ = _______
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118
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
2-5 Dividing Rational Numbers
Problem 1
3
4
FLIP 4
3
3
What is the reciprocal of 4?
1
1
3 4 3 4
• = • 1
4 3 4 3
1
1
Wow! The product is 1.
Problem 2
How do you make 0.4
a whole number?
74.8
7.48 10
4
0.4 10
0.4 (x) 4
x 10
Multiply by 10.
Think and Discuss
4
1
4
1. Given 5 8, explain what you will multiply 5 by to find the
quotient.
2. By what do you multiply the numerator and denominator of
12.62
in order to divide?
3
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28
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-6
Practice A
Solving Equations with Rational Numbers
Solve.
1. x + 1.2 = 4.6
2. a 3.4 = 5
_______________________
4.
x
=2
1.3
________________________
5. 6.7 + w = 1.1
_______________________
7. 7.2 = 0.9y
2
2
+x=
5
5
_______________________
13. x 3
1
=
2
5
_______________________
________________________
6.
________________________
8. k 4.05 = 6.2
_______________________
10. 3. 2.2m = 4.4
________________________
9.
________________________
11.
1
1
x=
4
2
12.
3
5
=
7
7
d
= 3.75
3.2
________________________
________________________
14. x n
= 3.8
1.9
1
3
a =
3
4
________________________
5
5
15. a =
6
8
________________________
________________________
3
1
in. high. The ceiling is 90 in. high.
4
2
How much higher is the ceiling than Elisa’s highest reach?
16. Elisa can reach 77
________________________________________________________________________________________
17. Nolan Makes $10.60 an hour at his after-school job. Last week
he worked 11.25 hr. How much was Nolan paid for the week?
________________________________________________________________________________________
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123
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-6
Reading Strategies
Follow a Procedure
The rules for solving equations with rational numbers are the same
as equations with whole numbers.
Get the variable
by itself.
Perform the same
operation on both
sides to keep the
equation balanced.
Follow the steps above to help you solve
Use the rules for
computing rational
numbers.
3
1
+y= .
4
4
1. What is the first step to solve this equation?
________________________________________________________________________________________
2. What operation should you use?
________________________________________________________________________________________
3. Write an equation to show the subtraction of
1
on both sides.
4
________________________________________________________________________________________
4. What is the value of y?
________________________________________________________________________________________
Follow the steps above to solve x – 4.5 = 13.
5. What is the first step to solve this equation?
________________________________________________________________________________________
6. What operation should you use?
________________________________________________________________________________________
7. Write an equation to show the addition of 4.5 to both sides.
________________________________________________________________________________________
8. Find the value of x.
________________________________________________________________________________________
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129
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class__________________
Review for Mastery
LESSON
2-6
Solving Equations with Rational Numbers
Solving equations with rational numbers is basically the same as
solving equations with integers or whole numbers:
Use inverse operations to isolate the variable.
1
z = 16
4
1
4 • z = 16 • 4
4
3.5
x
3.5
= 20.92
3
7
=
8
8
+
3
3
+
8
8
Multiply
each side
by 4.
z = 64
x + 3.5 = 17.42
y
=
y
3
to
8
each side.
Add
10
2
1
=1 =1
8
8
4
26t = 317.2
Subtract 3.5
from each side.
Divide each
side by 26.
26t = 317.2
26
26
t = 12.2
Tell what you would do to isolate the variable.
1. x 1.4 = 7.82
2.
________________________
1
7
+y=
4
4
________________________
3. 3z = 5
________________________
Solve each equation.
4. 14x = 129.5
________________________
7. x + 53.8 = 1.2
________________________
5.
1
y = 27
3
________________________
8. 25 =
1
k
5
________________________
6. 265.2 =
z
22.1
________________________
9. m 3
2
=
5
3
________________________
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126
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
2-6 Solving Equations with Rational Numbers
Problem 1
A.
4.2p 12.6
12.6
4.2p
4.2
4.2
Why do you do this step?
B.
1
4
x 9 9
1
1
4
to get the variable
by itself
1
x 9 9 9 9
Why do you do this step?
Problem 2
2
house
5
3 houses
1 day
d days s
d days • houses per day number of houses
2
5
2 5
5
d • • 3 • 5 2
2
15
d 2
1
d 7
2
d • 3
Think and Discuss
1. What is different about solving an addition equation with
fractions than solving an addition equation with integers?
2. What must you be careful of when solving a multiplication or
division equation with decimals?
Copyright © by Holt McDougal.
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30
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
2-6
Problem Solving
Solving Equations with Rational Numbers
Write the correct answer.
1. In the last 150 years, the average
height of men in America has
1
increased by
foot. Today,
3
American men have an average
7
feet. What was the
height of 5
12
average height of American men 150
years ago?
2. Jaime has a ribbon that is
1
in. long. If she cuts the ribbon
2
3
into pieces that are
in. long, into
4
how many pieces can she cut?
23
Solution:
Today’s height:
5
Ribbon length:
____________________
Piece length:
____________________
Number of pieces:
7
ft
12
23
1
Increase:
ft
3
7
1
or
Height 150 years ago: 5
12
3
7
4
3
1
5
=5
or 5
12 12
12
4
The average height of American men
1
150 years ago was 5 ft.
4
1
2
47
3
3
=
÷
4
4
=
47
•
2
=
4
47
•
3
2
1
=
1
94
= 31
3
3
31 pieces can be cut.
Choose the letter for the best answer.
4. The balance in Susan’s checking
account was $245.35. After the bank
deposited interest into the account,
her balance was $248.02. How much
interest did the bank pay?
3. Justin Gatlin won the Olympic 100-m
dash in 2004 with a time of 9.85
seconds. His time was 0.95 seconds
faster than Francis Jarvis who won
the 100-m dash in 1900. What was
Jarvis’ time?
A $1.01
A 8.95 seconds
B $2.67
B 10.65 seconds
C $3.95
C 10.80 seconds
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34
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-1
Practice A
Solving Two-Step Equations
Describe the operation performed on both sides of the equation
in steps 2 and 4.
3x + 2 = 11
1.
2.
x
1 + 1 = 2 + 1 _________________
4
x
= 1
4
x
4 = 4(1) _________________
4
3x + 2 2 = 11 2 _________________
3x = 9
3x
9
=
3
3
x
1 = 2
4
_________________
x=3
x = 4
Solve.
3. 2x + 3 = 9
_______________
7. 5y 2 = 28
_______________
4.
x
1=5
3
5. 3a + 4 = 7
_______________
6.
_______________
8. 2x 7 = 7
9.
_______________
w2
= 1
5
x+2
= 3
2
________________
10. 2r + 1 = 1
_______________
________________
Write and solve a two-step equation to answer the question.
11. Pearson rented a moving van for 1 day. The total rental charge is
$66.00. A daily rental costs $45.00 plus $0.25 per mile. How
many miles did he drive the van?
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
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393
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-1
Reading Strategies
Analyze Information
Break a problem into parts and analyze the information.
Jill has $8 in her pocket now. She had $20 when she left for
the movies. How much money did she spend?
Answer the questions in Exercises 1–4 to solve this problem.
1. How much money did Jill start with?
________________________________________________________________________________________
2. How much money does Jill have left?
________________________________________________________________________________________
3. What is the difference between these two amounts?
________________________________________________________________________________________
4. How much money did Jill spend?
________________________________________________________________________________________
Mark paid $45 at the music store for 3 CDs and a pack of
batteries, before tax. The batteries cost $6. How much did
Mark pay for each of the CDs?
Answer the questions in Exercises 5–9 to solve this problem.
5. How much did Mark spend at the music store?
________________________________________________________________________________________
6. How much did Mark spend on batteries?
________________________________________________________________________________________
7. What is the difference between these two amounts?
________________________________________________________________________________________
8. Since Mark paid $39 for CDs, divide $39 by 3.
________________________________________________________________________________________
9. How much did Mark pay for each CD?
________________________________________________________________________________________
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400
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-1
Review for Mastery
Solving Two-Step Equations
To solve an equation, it is important to first note how it is formed.
Then, work backward to undo each operation.
4z + 3 = 15
z
3=7
4
z+3
=7
4
The variable is
multiplied by 4 and
then 3 is added.
The variable is divided
by 4 and then 3 is
subtracted.
3 is added to the
variable and then the
result is divided by 4.
To solve, first
subtract 3 and
then divide by 4.
To solve, first add 3
and then multiply by 4.
To solve, multiply
by 4 and then
subtract 3.
Describe how each equation is formed.
Then, tell the steps needed to solve.
1. 3x 5 = 7
The variable is ___________________________ and then ___________________________.
To solve, first ___________________________ and then ___________________________.
2.
x
+5=7
3
The variable is ___________________________ and then ___________________________.
To solve, first ___________________________ and then ___________________________.
3.
x+5
=7
3
5 is ___________________________ and then the result is ___________________________.
To solve, first ___________________________ and then ___________________________.
4. 10 = 3x 2
The variable is ___________________________ and then ___________________________.
To solve, first ___________________________ and then ___________________________.
5. 10 =
x2
5
2 is ___________________________ the variable and then the result is ____________________.
To solve, first ___________________________ and then ___________________________.
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396
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Review for Mastery
LESSON
7-1
Solving Two-Step Equations (continued)
To isolate the variable, work backward using inverse operations.
The variable is multiplied by 2 and
then 3 is added.
2x + 3 = 11 To undo addition,
3
3
subtract 3.
2x
=8
2x
2
=
To undo multiplication,
8
divide by 2.
2
x=4
Check: Substitute 4 for x.
?
2(4) + 3 =
11
?
8+3=
11
The variable is divided by 2 and
then 3 is subtracted.
x
3 = 11
To undo subtraction,
2
+3 +3
add 3.
x
= 14
To undo division,
2
x
2•
= 2 • 14 multiply by 2.
2
x = 28
Check: Substitute 28 for x.
28
?
3=
11
2
?
14 3 =
11
11 = 11 11 = 11 Complete to solve and check each equation.
6. 3t + 7 = 19
To undo addition,
_____
____
subtract.
To undo multiplication,
3t
= ____
3t ÷ ____ = ____ ÷ ____ divide.
t
= ____
w
7=5
To undo subtraction,
7.
3
_____
w
3
____
= ____
_________________
Check:
add.
To undo division,
_____
•
z 3 = ____
_____
= ____
w
7=5
3
3
w
= ____ • 12
multiply.
3
w = ____
z3
8.
=8
To undo division,
2
z3
_____ •
= ____ • 8 multiply.
2
____
Check: 3t + 7 = 19
?
3(_____) + 7 =
19 Substitute for t.
?
_____ + 7 = 19
To undo subtraction,
add.
?
7=
5
Substitute.
?
7=
5
_________________
Check:
z3
=8
2
3 ?
=8
2
2
Substitute.
?
=
8
_____________
z = ____
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397
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
7-1 Solving Two-Step Equations
Problem 1
How many tickets did the family buy?
That's $3.25,
please.
Lucky’s TicketBuying Service
$52.00
Total Cost (Price of 1 Ticket • Number of Tickets) Service fee
52.00 3.25
9.75t
48.75 9.75t
48.75
9.75
9.75
t
9.75
5 t
3.25
3.25
Step 1 Subtract
3.25 from both
sides.
Step 2 Divide
both sides by
9.75.
Think and Discuss
1. What would the total cost have been in Problem 1 if the family
had purchased 7 tickets?
2. What operations did you use to solve the equation in Problem 1?
3. Why do you multiply 9.75 by t in Problem 1?
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90
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-3
Practice A
Solving Literal Equations for a Variable
Solve each equation for the given variable.
1. P = 3s for s
2. S = 2V for V
______________________________________
________________________________________
3. F = ma for m
4. I = Prt for r
______________________________________
________________________________________
5. a = b + c for b
6. h = 5k + 6 for k
______________________________________
7. A =
________________________________________
bh
for h
2
8. 2c + 3d = e + 5 for e
______________________________________
________________________________________
9. 2c + 3d = e + 5 for d
10. 2c + 3d = e + 5 for c
______________________________________
________________________________________
11. What would be the width of a rectangular poster if the area was
390 square inches and the length was 26 inches?
________________________________________________________________________________________
12. The formula S = 2B + 2 rh gives the surface area S of a cylinder,
where r is the radius and h is the height. Solve this equation for r.
________________________________________________________________________________________
13. The formula A = P + Prt gives the amount A in an account earning
simple interest, where P is the principal, r is the rate, and t is the
time in years. If the amount is $484 on a principal amount of $400 at
a rate of 0.07, then for how long was the money saved?
________________________________________________________________________________________
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410
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-3
Reading Strategies
Use a Concept Map
Some equations are literal equations.
Examples
Definition
P = 4s
2w – 3 + x = 4z
An equation with two or more variables
A=
How to rearrange
Literal Equation
1
bh
2
When to rearrange
When using a formula to find the value of a
Use inverse operations
variable that is not isolated
--addition and subtraction undo each other
--multiplication and division undo each other
Answer each question.
1. Is the equation b + 3b = 15 a literal equation? Why or why not?
________________________________________________________________________________________
2. The formula P = 4s gives the perimeter P of a square where s is the
length of a side.
• Luis knows the side length and wants to know the perimeter of the square.
• Faye knows the perimeter and wants to know the length of each side.
Who should rearrange the formula to find what they need? ___________________
3. Which operation will isolate s in P = 4s? __________________________________
4. Solve P = 4s for s. ___________________________________________________
5. Consider the equation 2w – 3 + x = 4z. Locate the variable x.
What other terms are on the same side of the equation as x?
How can you isolate x?
_______________________
_______________________________________________________________
6. Solve 2w – 3 + x = 4z for x. ____________________________________________
1
7. Tell how you would solve A = bh for h. _________________________________
2
________________________________________________________________________________________
8. The area of a triangle is 80 sq cm and the base is 10 cm.
1
Solve A = bh for h. ___________ Find the height of the triangle. _____________
2
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416
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Review for Mastery
LESSON
7-3
Solving Literal Equations for a Variable
A literal equation has more than one variable. Solve literal equations the same way
you solve other equations—by using inverse operations.
Compare solving a = 3b + c for b as is, to solving it when a = 17 and c = 5.
a = 3b + c
a = 3b + c
a
= 3b + c
c
c
Step
17 = 3b + 5
Locate b.
17 = 3b + 5
Isolate 3b by subtracting.
a c = 3b
a c 3b
=
3
3
ac
=b
3
17 = 3b + 5
5
5
12 = 3b
12 3b
=
3
3
4=b
Isolate b by dividing.
ac
= b for a. Multiply each side by 3:
3
a – c = 3b. Then add c to each side: a = 3b + c. It is the original equation.
You can check your work by solving
Fill in the blanks to solve each equation for the given variable.
1. R = mt – 2 for t
2. k + 4m – g = p + 9m for k
Add _____ to both sides.
__________ 4m from both sides.
R + _____ = ______
k – g = p + ______
Divide both sides by _____.
R+
Add ____ to both sides.
=t
k = p + _____ + ____
4. z + 1 = 5 + y x for y
3. d + 2p = m for p
______________________________________
5. A =
________________________________________
m+n
for m
2
6. 15t = 12t + rs for s
______________________________________
________________________________________
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413
Holt McDougal Mathematics
Name
Date
LESSON
Class
Student Worksheet
7-3 Solving Literal Equations for a Variable
When you solve a literal equation, think how you would solve a
normal equation that looks like the literal equation.
Problem 1
Solve d r • t for r.
I’d solve
30 r • 6 by dividing
both sides by 6.
The equation looks like
this one: 30 r • 6
Solving 30 r • 6 for r
30 r • 6
30 r • 6
6
6
Solving d r • t for r
Here you divide
both sides by 6...
5r
dr•t
d r •t
t
t
d
r
t
...so here, divide
both sides by t.
Problem 2
Solve y 3x b for b.
I’d solve
12 9 b by subtracting
9 from both sides.
The equation looks like
this one: 12 9 b.
Solving 12 9 b for b
Here you subtract
9 both sides...
Solving y 3x b for b
12 9 b
12 9 9 b 9
3b
y 3x b
...so here, subtract 3x
from both sides.
y 3x 3x b 3x
y 3x b
Think and Discuss
1. Why might you want to solve the formula d rt for r?
2. Show how you would solve the circumference formula C 2πr
for r. Explain each step.
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94
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-3
Problem Solving
Solving Literal Equations for a Variable
Write the correct answer.
1. The formula 3F 24 = s is used to
find the shoe size of an adult whose
foot is F inches long. Solve the
equation for F. How long is an adult’s
foot if their shoe size is 6?
2. The formula 4t – 148 = c gives the
number of times a cricket chirps c, in
one minute, when the temperature is t
(in °F). Solve the equation for t. Find
the temperature when a cricket chirps
100 times per minute.
Solution:
Solution:
Solve for F.
3F 24 = s
3F 24 + 24 = s + 24
3F = s + 24
3F
s + 24
=
3
3
s + 24
F=
3
Find the length of a size 6 foot.
6 + 24
F=
3
30
F=
3
F = 10 inches
4t 148 = c
4t 148 + ___ = c + ___
4t = c + ___
___ = c + ___
t = c + ___
Find the temperature when a cricket
chirps 100 times per minute.
t = ___ + ___
t = ___
t = ___
4. Euler’s formula states that the number
of vertices V in a polyhedron is equal
to 2 plus the number of edges E
minus the number of faces F. This is
written as V = 2 + E F. Solve the
formula for E. How many edges does
a polyhedron have if it has 6 vertices
and 5 faces?
r 2h
gives the
3
volume V of a cone with radius r and
height h. Solve the equation for h.
Find the height of an ice cream cone if
its volume is 19.2325 cubic inches
and its radius is 1.75 inches.
Use 3.14 for .
3. The formula V =
________________________________________
_______________________________________
Choose the letter of the best answer.
5. The formula v = 1053.52 + 1.14t
gives the speed v at which sound
travels in feet per second when the
air temperature is t (in °F). What is
the air temperature when sound
travels at 1139.02 ft/sec?
A 54°F
6. The formula V = lwh gives the
volume V of a rectangular prism,
with length l, width w, and height h.
Find the length of a crate with a
volume of 87.75 cubic feet, width of
4.5 feet, and height of 3 feet.
A 6.5 ft
C 75°F
C 7.3 ft
B 11.7 ft
B 61°F
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118
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Name _______________________________________ Date __________________ Class __________________
LESSON
7-4
Practice A
Solving Inequalities by Adding or Subtracting
Compare. Write < or >.
1. 3 + 8 ___ 12
2. 5(3) ___ 14
3. 15 7 ___ 7
4. 3(7) ___ 28
5. 9 + (9) ___ 16
6. 10(6) ___ 65
7. 4 9 ___ 12
8. 3 + 6 ___ 8
9. 7(8) ___ 50
Solve and graph each inequality.
10. a + 3 < 7
_______________________
13. 3 + s > 1
_______________________
16. g + 2 2
_______________________
19. t 4 < 7
_______________________
22. x + 2 > 1
_______________________
11. 4 + m 1
________________________
14. z 5 5
________________________
17. 4 + w 3
________________________
20. 6 + r 5
12. n 1 < 2
________________________
15. 9 + p < 14
________________________
18. k 1 < 5
________________________
21. y + 14 > 22
________________________
23. 4 + a < 6
________________________
________________________
24. c 2 3
________________________
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418
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-4
Reading Strategies
Reading a Table
The word inequality means not equal. Inequality symbols are used
to compare values that are not equal. The table shows the inequality
symbols and their meanings.
Inequality Symbols
Symbol
>
<
Meaning
Greater than
Less than
Greater than or equal to
Less than or equal to
Use the table to help you write the correct symbol.
1. Write the symbol you use to show that
one number is greater than another.
2. Write the symbol you use to show that
a number is less than or equal to
another number.
_______________________________________
________________________________________
3. Write the symbol you use to show that
one number is greater than or equal
to another
4. Write the symbol you use to show that
one number is less than another.
________________________________________
_______________________________________
An inequality that includes a variable is called an algebraic
inequality.
Study the word phrases and symbols for algebraic inequalities.
Word Phase
z is less than negative five
y is less than or equal to twelve
m is greater than twenty-three
t is greater than or equal to ten
Symbols
z < 5
y 12
m > 23
t 10
5. Write the symbols for “n is less than 12.”
________________________________________________________________________________________
6. Write the word phrase for “v 8.”
________________________________________________________________________________________
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424
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Review for Mastery
LESSON
7-4
Solving Inequalities by Adding or Subtracting
A solution of an inequality is a number that makes the inequality
true. An inequality usually has more than one solution. All the
solutions are contained in the solution set.
As with equations, solve a simple inequality by using inverse
operations to isolate the variable.
Solve and graph x + 4 > 9.
x+4>9
4 4
x>5
Draw an open circle at 5 to
show that 5 is not included in
the solution set.
Draw an arrow to the right
of 5 to show that all
numbers greater than 5 are
included in the solutions.
Subtract 4.
According to the graph, 6 should be a
solution and 4 should not be a solution.
Check:
x+4>9
x+4>9
?
6+4 > 9
10 > 9
?
4+4 > 9
8 >/ 9
So, 6 is in the solution set and 4 is not in the solution set.
Thus, the solution set for the inequality x + 4 > 9 is x > 5.
Write true or false.
1. 7 < 4
3. 3 > 4
2. 0 9
_______________________
________________________
________________________
Using the variable n, write the inequality shown by each graph.
4.
5.
_________________
_________________
Complete. Is the given value in the solution set? Answer is or is not.
6. 3 ___ in the solution
7. 0 ___ in the solution
8. 14 __ in the solution
set of x 1 > 5.
x1>5
____
?
1 > 5
__________
set of z + (4) 4.
z + (4) 4
____
+ (4) 4
>5
____
4
set of w + 10 25.
w + 10 25
____
+ 10 25
____
25
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421
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Name
LESSON
Date
Class
Student Worksheet
7-4 Solving Inequalities by Adding or Subtracting
8
5
5
ab
a is greater than b.
8
ab
a is less than b.
Problem 1
Completing an Inequality
Look for operations
Simplify
Then complete the inequality
13 9 ? 6
4 ? 6
4 6
This is an operation.
Problem 2
⫺19
⫺18
⫺17
⫺18
⫺16
⫺15
⫺17
Think of an open circle as a hole in the graph. Because of the hole,
the graph does not touch 17. So 17 is not a solution.
Think and Discuss
1. How would you complete the inequality in Problem 1 if the left
side were 12 7?
2. Write in words what the graph in Problem 2 shows.
3. Is 17 a solution to Problem 2?
Is 18.9?
4. In Problem 2, if the circle was solid at point 17, what would
that mean?
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96
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-5
Practice A
Solving Inequalities by Multiplying or Dividing
Solve and graph
2. 3 1. 4x > 20
_______________
3. _______________
b
3
8
4. 6d < 18
_______________
_______________
6. 5. 63 7f
g
2
4
_______________
_______________
7. 13 <
y
5
h
3
8. 7j > 14
_______________
_______________
9. Cheryl wants to buy a bicycle that costs $160. If she saves $12 each week, what is
the fewest number of weeks she must save in order to buy the bicycle?
________________________________________________________________________________________
1
the amount of time that his brother
3
did. If Mark spent 25 minutes on his math homework, how much time did his
brother spend on his math homework?
10. Mark worked on math homework less than
________________________________________________________________________________________
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426
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-5
Reading Strategies
Understand Symbols
If you know the meanings of the inequality symbols, you can read
and write inequalities as word sentences, and you can write word
sentences as inequalities.
< less than
> greater than, or more than
less than or equal to, or no more than
greater than or equal to, or at least
Inequality
6<x
y > 14
15 z
b5
Word Sentence
Six is less than x.
y is greater than fourteen, or y is
more than fourteen.
Fifteen is less than or equal to z, or fifteen
is no more than z.
b is greater than or equal to five, or b is at least 5.
Many inequalities include multiplication or division.
Inequality
Word Sentence
21 < 3x
Twenty-one is less than three times x.
y
>8
y divided by three is greater than eight.
3
4 2z
Four is less than or equal to two times z,
or four is no more than 2z.
b
10
b divided by four is greater than or equal to ten,
4
or b divided by four is at least ten.
Write the inequality as a word sentence.
1. 5d > 40
________________________________________________________________________________________
2.
f
3 _________________________________________________________________________________
6
________________________________________________________________________________________
3. 11 <
g
2
________________________________________________________________________________
4. 16 4h ________________________________________________________________________________
________________________________________________________________________________________
Write an inequality that you could use to solve the problem.
5. A tree is more than five times as tall as a math student. The tree is 28 feet tall.
________________________________________________________________________________________
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432
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class__________________
LESSON
7-5
Review for Mastery
Solving Inequalities by Multiplying or Dividing
To solve an inequality, multiply and divide the same way you would
solve an equation. But, if you multiply or divide by a negative
number, you must reverse the inequality sign.
Divide by a Positive Number
2x < 14
2x 14
<
2
2
x<7
Divide by a Negative Number
2x < 14
2x 14
>
Reverse the inequality sign.
2
2
x > 7
To check your solution, choose two
numbers from the graph and substitute
them into the original equation.
Choose a number that should be a solution
and a number that should not be a solution.
Check
According to the graph, 6 should
be a solution, but 8 should not be.
2x < 14
?
2 i 8 < 14
16 </ 14
Complete to solve. Then graph the equation and check.
s
1. 3y 24
2.
<4
9
s
24
3y
_____ •
_____
9
3
y ________
s ________
________
2x < 14
?
2 i 6 < 14
12 < 14
_____ _____
•4
________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
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429
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
7-5 Solving Inequalities by Multiplying or Dividing
Problem 1
Problem 2
Remember
Remember
An open circle means the point on
the graph is not part of the solution.
Use the symbols or .
A closed circle means the point on
the graph is part of the solution. Use
the symbols or .
Solve and graph.
Solve and graph.
h
24 5
7x 42
h
5 • 24 is 120 and 5 • 5 is h
Think:
The number 7 is negative,
reverse to .
7x
x
7
42
6
7
120 h, or h 120
Divide.
Divide.
x 6
115 116 117 118 119 120 121 122
The circle is open because h cannot be
120. It is not part of the solution.
12 11 10 9 8 7 6 5 4
The circle is closed because 6 is part of
the solution.
All the numbers less than 120 are part of
the solution.
The number 6 and the negative
numbers less than 6 are part of the
solution.
Think and Discuss
1. What does the symbol represent? What does the symbol represent?
2. When graphing the solution of an inequality, when do you use an
open circle? When do you use a closed circle?
3. Why did you have to flip the inequality sign in Problem 2?
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98
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-6
Practice A
Solving Two-Step Inequalities
Write yes or no to tell whether the inequality symbol would be
reversed in the solution. Do not solve.
1. 2x – 4 < 20
2. 4 3y 21
_______________
3. 6x + 17 > 3
_______________
4. _______________
a
4 2
5
________________
Solve.
5. 2x – 17 29
6. 8 _______________
9. 10 10 2d 5
_______________
k
< 12
2
7. 23 3w < 34
_______________
10.
8. 24 0.6x < 60
_______________
2x
+ 5 14
3
11.
_______________
2
y
1
3
6
2
_______________
________________
12.
a
1
1
+
>
7
7
14
________________
Solve and graph.
13. 2x 1 < 3
14. 16 1 – 3a
_______________________________________
15.
________________________________________
y
3
1
2
4
2
16.
_______________________________________
d
5
1
+
>
3
12
4
________________________________________
17. Mrs. Ocosta is paid a 5% commission on her sales each week.
In addition, she receives a base salary of $375. What should
the amount of her sales be for the week if she hopes to make
at least $600 this week?
________________________________________________________________________________________
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434
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
LESSON
7-6
Reading Strategies
Follow a Procedure
You can use these steps to help you solve a two-step inequality.
Solve 8 < 4x + 4.
Step 1: Get the variable
by itself on one
8
4 < 4x + 4 4
Subtract 4 from both sides.
side of the inequality. 12 < 4x
Step 2: Solve.
4x
12
<
4
4
3
Step 3: Rewrite the solution
so the variable
comes first.
Divide both sides by 4.
<x
x > 3
Use the procedure to answer each question.
1. What did the procedure tell you to do first?
________________________________________________________________________________________
2. How did you get the variable by itself in this problem?
________________________________________________________________________________________
3. What is the second step given?
________________________________________________________________________________________
4. How did you solve this inequality?
________________________________________________________________________________________
5. How would the graph for x 3 be different than the above graph?
________________________________________________________________________________________
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441
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Review for Mastery
LESSON
7-6
Solving Two-Step Inequalities
To solve an inequality, undo operations the same way you would
with an equation. But, when multiplying or dividing by a negative
number, reverse the inequality symbol.
3x + 2 > 11 To undo addition,
2
3x
3x + 2 > 11 To undo addition,
2 subtract 2.
2 2
3x
> 9 To undo multiplication,
subtract 2.
>9
To undo multiplication,
3x
9
<
3
3
3x
9
>
divide by 3.
3
3
divide by 3 and
x < 3 change > to <.
x >3
The solution set contains all real
numbers greater than 3.
The solution set contains all real
numbers less than 3.
Complete to solve and graph.
1. 2t + 1 9
______
2t
___
_____
2t
____
t
2. 2t + 1 9
To undo addition,
subtract.
______
To undo multiplication.
2t
8
divide.
2t
___
3. 3z 2 > 1
________
3z
3z
z
>
_______
3z >
________
3z
_____
To undo multiplication.
divide by 2 and
________
4. 3z 2 >
________
subtract.
___
t
________
To undo addition,
change to .
1
________
________
_____
z __________
_________
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437
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Review for Mastery
LESSON
7-6
Solving Two-Step Inequalities (continued)
To solve multistep inequalities, you may need to clear fractions.
Multiply both sides by the LCD.
v
1
1
+
>
The LCD is 4.
4
4
2
v
1
1
4•
+4•
>4•
Multiply by the LCD.
4
4
2
v + 1 > 2
1
1
Subtract from both sides.
v > 3
Complete to solve and graph.
5.
________
•
b
7
2
4 12
3
Find the LCD.
b
7
2
________ •
________ •
4
12
3
________
Multiply by the LCD.
b ________ ________
________
________
Add.
________
b ________
b
Divide and change symbol.
__
b ________
y
2
1
+
7 14
2
6.
_____
•
Check direction.
7.
y
2
1
+ _____ •
_____ • 14
7
2
_____
•
x
1
2
>
+
9
3
3
x
1
2
> _____ •
+ _____ •
9
3
3
________
+ ________ ________
________
> x + ________
________
________
________
________
________
________
________
>x
x _____ _________
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438
Holt McDougal Mathematics
Name
LESSON
Date
Class
Student Worksheet
7-6 Solving Two-Step Inequalities
Problem 1
Solve and graph.
7y 4 . 24
4
y is multiplied
by 7. Divide
to undo.
4
7y . 28
4 is subtracted
from 7y. Add 4 to
undo.
7y
28
.
7
7
y can be any number
to the right of 4 on
the number line.
y .4
2
0
2
4
6
Problem 2
To break even, ticket sales plus money in budget must be greater
than cost of production.
REVENUE
COST
Price of the
ticket ($4.75).
This is money
the club must
spend.
R C
This is money club
has in budget and
earns from ticket
sales.
Number of
tickets sold (t).
4.75t 610.75
Amount from
fundraising ($610.75)
Cost of entire
production
1100.00
Think and Discuss
1. List 3 values that are solutions to the inequality in Problem 1.
2. What can the drama club do in Problem 2 to make sure that
they earn money on their musical?
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