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Transcript
Chapter 1
A1
D
D
A
A
D
A
D
D
5. The phrase twice the sum of a number squared and 6 could
be written as 2n2 6.
6. The reflexive property of equality states that if a b then
b a.
7. The absolute value of a number is its distance from 0 on
the number line.
8. If the absolute value of any expression is equal to a negative
number, then the solution is the empty set.
9. When adding or subtracting a negative number to both sides
of an inequality, the inequality symbol must be reversed.
10. Writing a solution in the form {x| x 5} is called set builder
notation.
11. If a compound inequality contains the word “or”, the solution
will be the intersection of the solution sets of the
two inequalities.
12. If |3x 1| 10, then 3x 1 10 and 3a 1 10.
After you complete Chapter 1
A
4. The commutative property is true for addition and
multiplication only.
A
Chapter 1
3
Glencoe Algebra 2
• For those statements that you mark with a D, use a piece of paper to write an example of
why you disagree.
• Did any of your opinions about the statements change from the first column?
time
drops
minutes
drops
.
.
per milliliter.
milliliters
per minute.
Chapter 1
5
Answers
Glencoe Algebra 2
Sample answer: Please excuse my dear Aunt Sally. (parentheses;
exponents; multiplication and division; addition and subtraction)
4. Think of a phrase or sentence to help you remember the order of operations.
Remember What You learned
operations, different people might get different answers.
3. Why is it important for everyone to use the same order of operations for evaluating
expressions? Sample answer: If everyone did not use the same order of
Multiply the difference of 13 and 5 by the sum of 9 and 21. Add the result to 10. Then
divide what you get by 2. [(13 5)(9 21) 10] 2
2. Read the following instructions. Then use grouping symbols to show how the instructions
can be put in the form of a mathematical expression.
c. {14 [8 (3 12)2]} (63 100) (3 12)
b. 9 [5(8 6) 2(10 7)] (8 6) and (10 7)
a. [(3 22) 8] 4 (3 22)
1. There is a customary order for grouping symbols. Brackets are used outside of
parentheses. Braces are used outside of brackets. Identify the innermost expression(s) in
each of the following expressions.
Read the Lesson
8 60
and is measured in
of solution and is measured in
and is measured in
and is measured in
drop factor
volume
flow rate
• Write the expression that a nurse would use to calculate the flow rate of an IV if a doctor
orders 1350 milliliters of IV saline to be given over 8 hours, with a drop factor of 20 drops
per milliliter. Do not find the value of this expression. 1350 20
t represents
d represents the
F represents the
A
D
The order of operations must be followed so that every
expression will have only one value.
Chapter Resources
V represents the
3. All real numbers are in the set of rational numbers.
2.
1. Algebraic expressions contain at least one variable.
Statement
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
STEP 2
A or D
Answers
• Reread each statement and complete the last column by entering an A or a D.
STEP 2
STEP 1
A, D, or NS
the quantity that each of the variables in this formula represents and the units in which
each is measured.
• Nurses use the formula F to control the flow rate for IVs. Name
t
Vd
Read the introduction to Lesson 1-1 in your textbook.
____________ PERIOD _____
8:03 AM
• Write A or D in the first column OR if you are not sure whether you agree or disagree,
write NS (Not Sure).
• Decide whether you Agree (A) or Disagree (D) with the statement.
Expressions and Formulas
Lesson Reading Guide
Get Ready for the Lesson
1-1
NAME ______________________________________________ DATE
5/19/06
• Read each statement.
Before you begin Chapter 1
Equations and Inequalities
Anticipation Guide
____________ PERIOD _____
A2-01-873971
STEP 1
1
NAME ______________________________________________ DATE
Lesson 1-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A1
(Lesson 1-1)
Glencoe Algebra 2
Chapter 1
Expressions and Formulas
Study Guide and Intervention
5. (5 23)2 52 144
8. (7 32)2 62 40
4. 9(32 6) 135
A2
7. 6
2
6
26. b c 4 d 15
25. a b c 7.45
Chapter 1
23. cd 4
22. 3 b 21
b
d
20. (b c)2 4a 81.8
19. ac bd 31.3
dc 17. 5(6c 8b 10d) 215
16. 49.2
ab
d
Glencoe Algebra 2
27. d 8.7
a
bc
24. d(a c) 6.1
21. 6b 5c 54.4
a
d
18. 6
c2 1
bd
Evaluate each expression if a 8.2, b 3, c 4, and d .
1
2
13. 8(42 8 32) 240
4
6 9 3 15
15. 82
642
14. 461
24
12. 6(7) 4 4 5 38
11. 14 (8 20 2) 7
10. 12 6 3 2(4) 6
9. 20 22 6 11
6. 52 18 2 34.25
1
4
3. 2 (4 2)3 6 4
Example
np
500
To calculate the number of reams of paper needed to print n
(172)(25)
500
43,000
500
Chapter 1
7
Glencoe Algebra 2
4. A person’s basal metabolic rate (or BMR) is the number of calories needed to support his
or her bodily functions for one day. The BMR of an 80-year-old man is given by the
formula BMR 12w (0.02)(6)12w, where w is the man’s weight in pounds. What is the
BMR of an 80-year-old man who weighs 170 pounds? 1795 calories
3. Sarah takes 40 breaths to blow up the beach ball. What is the average volume of air per
breath? about 1250 cm3
sphere and r is its radius. What is the volume of the beach ball in cubic centimeters?
(Use 3.14 for .) 50,015 cm3
2. The volume of a sphere is given by the formula V r3, where V is the volume of the
4
3
1. Her beach ball has a radius of 9 inches. First she converts the radius to centimeters
using the formula C 2.54I, where C is a length in centimeters and I is the same length
in inches. How many centimeters are there in 9 inches? 22.86 cm
For a science experiment, Sarah counts the number of breaths needed for her to blow up a
beach ball. She will then find the volume of the beach ball in cubic centimeters and divide
by the number of breaths to find the average volume of air per breath.
For Exercises 1–3, use the following information.
Exercises
You cannot buy 8.6 reams of paper. You will need to buy 9 reams to print 172 copies.
8.6
r np
500
Substitute n 172 and p 25 into the formula r .
is the number of reams needed. How many reams of paper must you buy to print
172 copies of a 25-page booklet?
copies of a booklet that is p pages long, you can use the formula r , where r
Answers
16 23 4
12
2. 11 (3 2)2 14
1. 14 (6 2) 17
Find the value of each expression.
Replace each variable with the given value.
3x2 x(y 5) 3 (3)2 3(0.5 5)
3 (9) 3(4.5)
27 13.5
13.5
Evaluate 3x2 x(y 5) if x 3 and
y 0.5.
(continued)
Formulas A formula is a mathematical sentence that uses variables to express the
relationship between certain quantities. If you know the value of every variable except one
in a formula, you can use substitution and the order of operations to find the value of the
unknown variable.
Expressions and Formulas
Study Guide and Intervention
____________ PERIOD _____
8:03 AM
Exercises
[18 (6 4)] 2 [18 10] 2
82
4
Evaluate [18 (6 4)] 2.
Example 2
Simplify the expressions inside grouping symbols.
Evaluate all powers.
Do all multiplications and divisions from left to right.
Do all additions and subtractions from left to right.
1-1
NAME ______________________________________________ DATE
5/19/06
Example 1
Order of
Operations
1.
2.
3.
4.
____________ PERIOD _____
A2-01-873971
Order of Operations
1-1
NAME ______________________________________________ DATE
Lesson 1-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A2
(Lesson 1-1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Expressions and Formulas
Skills Practice
6(7 5)
6. 4
1
5. [9 10(3)]
3
1
2
8. [3(5) 128 22]5 85
A3
25
2
rv3
s
20. 7s 2v 22
19. 9r2 (s2 1)t 105
Chapter 1
8
Glencoe Algebra 2
for a given temperature in degrees Fahrenheit. What is the temperature in degrees
Celsius when the temperature is 68 degrees Fahrenheit? 20C
5
9
22. TEMPERATURE The formula C (F 32) gives the temperature in degrees Celsius
21. TEMPERATURE The formula K C 273 gives the temperature in kelvins (K) for a
given temperature in degrees Celsius. What is the temperature in kelvins when the
temperature is 55 degrees Celsius? 328 K
2w
r
18. 0
2
17. w[t (t r)] 16. 4
3v t
5s t
14. s2r wt 3
13. (4s)2 144
8(13 37)
3
4
(8)2
9
5
c
e
22. 2ab2 (d 3 c) 67
ac4
d
20. 2 22
18. ac3 b2de 70
d(b c)
16. 12
ac
14. (c d)b 8
Chapter 1
9
Answers
Glencoe Algebra 2
25. AGRICULTURE Faith owns an organic apple orchard. From her experience the last few
seasons, she has developed the formula P 20x 0.01x2 240 to predict her profit P in
dollars this season if her trees produce x bushels of apples. What is Faith’s predicted
profit this season if her orchard produces 300 bushels of apples? $4860
24. PHYSICS The formula h 120t 16t2 gives the height h in feet of an object t seconds
after it is shot upward from Earth’s surface with an initial velocity of 120 feet per
second. What will the height of the object be after 6 seconds? 144 ft
Fahrenheit for a given temperature in degrees Celsius. What is the temperature in
degrees Fahrenheit when the temperature is 40 degrees Celsius? 40F
23. TEMPERATURE The formula F C 32 gives the temperature in degrees
21. 9bc 141
1
e
19. b[a (c d) 2 ] 206
17. (b de)e2 1
ab
c
15. d2 12
13. ab2 d 45
1
3
12. (1)2 4(9) 53
59
Evaluate each expression if a , b 8, c 2, d 3, and e .
11. 32
6
1
4
10. [5 5(3)] 5
8. [4(5 3) 2(4 8)] 16 1
7. 18 {5 [34 (17 11)]} 41
1
2
6. (2)3 (3)(8) (5)(10) 18
5. 20 (5 3) 52(3) 85
9. [6 42] 5
4. 12 [20 2(62 3 22)] 88
3. 1 2 3(4) 2 3
Answers
15. 2(3r w) 7
12. s 2r 16v 1
10. 2st 4rs 84
11. w(s r) 2
9. 6r 2s 0
Evaluate each expression if r 1, s 3, t 12, v 0, and w .
7. (168 7)32 43 152
3
4. 5 3(2 12 2) 7
3. (3 8)2(4) 3 97
2. 4(12 42) 16
1. 3(4 7) 11 20
____________ PERIOD _____
8:03 AM
7
2. 9 6 2 1 13
Expressions and Formulas
Practice
Find the value of each expression.
1-1
NAME ______________________________________________ DATE
5/19/06
1. 18 2 3 27
____________ PERIOD _____
A2-01-873971
Find the value of each expression.
1-1
NAME ______________________________________________ DATE
Lesson 1-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A3
(Lesson 1-1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
R
A4
Chapter 1
subtraction
10 + 3(4 + 6) = 4
10
17 min
Glencoe Algebra 2
6. Use your formula to compute the number
of minutes it would take to broil a 2 inch
thick steak.
Sample Answer: T 5(w 1) 12
or T 5w 7
5. Write a formula for T in terms of w.
5.7
5.0
4.4
4.0
3.6
3.3
3.0
2.8
2.6
Time (hours)
2
Chapter 1
11
Glencoe Algebra 2
You can make it to your aunt’s house on time, but won’t have enough
gas to get home.
the speeds needed to get to your aunt’s. Will you make it?
30
1
5
formula M S2 S, where S is your speed. Determine your fuel rate for
4. Cost depends on the cost of gasoline, the number of total miles of the trip, and
your car’s fuel efficiency (mi/gal). The miles per gallon can be found using the
About 8.5 gallons
3. How many gallons of gas can you buy?
Now determine if you can afford enough gasoline to make the trip and return.
You will miss the party for all speeds 50 mph and less, assuming an
11 A.M. departure.
2. For which speed(s), will you miss the surprise birthday party?
75
70
65
60
55
50
45
40
35
Speed (mph)
and S is the speed. Determine travel time to your aunt’s house at various speeds.
S
D
miles per hour (mph), T , where T is time in hours, D is the distance (200 miles),
1. A simple formula relates the travel time, depending on your average speed in
Answers
3. GUESS AND CHECK Amanda
received a worksheet from her teacher.
Unfortunately, one of the operations in an
equation was covered by a blot. What
operation is hidden by the blot?
236 in2
r
A steak has thickness w inches. Let T be
the time it takes to broil the steak. It takes
12 minutes to broil a one inch thick steak.
For every additional inch of thickness,
the steak should be broiled for 5 more
minutes.
the following information.
COOKING For Exercises 5 and 6, use
56 mi
First determine at which speed you must travel to arrive by 3:00.
8:03 AM
2. GEOMETRY The formula for the area
of a ring-shaped object is given by A (R2 r2), where R is the radius of the
outer circle and r is the radius of the
inner circle. If R 10 inches and r 5
inches, what is the area rounded to the
nearest square inch?
10 6 10 12 or 10(6 12)
Traveling on a Budget
Enrichment
____________ PERIOD _____
You are traveling to your aunt’s house 200 miles away for a surprise birthday party.
The party starts at 3 P.M. but you cannot leave from your house before 11 A.M. You
must fill your gas tank before the trip. Gasoline is $3.50 per gallon and you have $30.
Will you make it to the party and make it back home?
1-1
NAME ______________________________________________ DATE
5/19/06
number of miles that Rick can drive on g
gallons of gasoline is given by m 21g.
How many miles can Rick drive on $8
worth of gasoline?
3
d
is given by g . The formula for the
4. GAS MILEAGE Rick has d dollars. The
formula for the number of gallons of
gasoline that Rick can buy with d dollars
Expressions and Formulas
Word Problem Practice
____________ PERIOD _____
A2-01-873971
1. ARRANGEMENTS The chairs in an
auditorium are arranged into two
rectangles. Both rectangles are 10 rows
deep. One rectangle has 6 chairs per
row and the other has 12 chairs per row.
Write an expression for the total number
of chairs in the auditorium.
1-1
NAME ______________________________________________ DATE
Lesson 1-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A4
(Lesson 1-1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Properties of Real Numbers
Lesson Reading Guide
A5
Chapter 1
12
Glencoe Algebra 2
A commuter is someone who travels back and forth to work or another
place, and the commutative property says you can switch the order when
two numbers that are being added or multiplied. An association is a
group of people who are connected or united, and the associative
property says that you can switch the grouping when three numbers are
added or multiplied.
4. How can the meanings of the words commuter and association help you to remember the
difference between the commutative and associative properties? Sample answer:
Remember What You Learned
The quantities a, b, and c are used in the same order, but they are grouped
differently on the two sides of the equation. The second equation uses the
quantities in different orders on the two sides of the equation. So the
second equation uses the Commutative Property of Multiplication.
3. Consider the equations (a b) c a (b c) and (a b) c c (a b). One of the
equations uses the Associative Property of Multiplication and one uses the Commutative
Property of Multiplication. How can you tell which property is being used in each
equation? The first equation uses the Associative Property of Multiplication.
natural numbers
whole numbers
integers
N
W
Z
{all nonterminating, nonrepeating decimals}
{…, 3, 2, 1, 0, 1, 2, 3, …}
{0, 1, 2, 3, 4, 5, 6, 7, 8, …}
{1, 2, 3, 4, 5, 6, 7, 8, 9, …}
Exercises
6
7
15
3
13
Glencoe Algebra 2
20. 0.02 Q, R
17. 2 I, R
5
14. 42
I, R
11. 4.1
7
Q, R
8. 26.1 Q, R
Answers
19. 894,000 N, W, Z, Q, R
18. 33.3
Q, R
Chapter 1
16. Q, R
8
13
13. 1 Z, Q, R
10. N, W, Z, Q, R
36
7. Q, R
9
3. 0 W, Z, Q, R
15. 11.2 Q, R
1
2
6. 34 Q, R
2. 81
Z, Q, R
12. N, W, Z, Q, R
5
25
9. I, R
5. 73 N, W, Z, Q, R
1. Q, R
4. 192.0005 Q, R
naturals (N), wholes (W), integers (Z), rationals (Q), reals (R)
rationals (Q), reals (R)
25
25
5
m
Name the sets of numbers to which each number belongs.
b.
11
a. 3
{all rationals and irrationals}
{all numbers that can be represented in the form , where m and n are integers and
n
n is not equal to 0}
Name the sets of numbers to which each number belongs.
irrational numbers
I
Example
rational numbers
Q
Answers
Sample answer: (12 18) 45 30 45 75;
12 (18 45) 12 63 75
2. Write the Associative Property of Addition in symbols. Then illustrate this property by
finding the sum 12 18 45 in two different ways. (a b) c a (b c);
decimal because there is a block of digits, 57, that repeats forever, so
this number is rational. The number 0.010010001… is a non-repeating
decimal because, although the digits follow a pattern, there is no block
of digits that repeats. So this number is an irrational number.
1. Refer to the Key Concepts box on page 11. The numbers 2.5
7
and 0.010010001… both
involve decimals that “go on forever.” Explain why one of these numbers is rational and
the other is irrational. Sample answer: 2.5
7
2.5757… is a repeating
Read the Lesson
coupon amounts or add the amounts for the scanned coupons and
multiply the sum by 2.
real numbers
R
8:03 AM
• Describe two ways of calculating the amount of money you saved by using coupons if your
register slip is the one shown on page 11. Sample answer: Add all the individual
from the total amount of purchases so that you save money by using
coupons.
• Why are all of the amounts listed on the register slip at the top of the page followed by
negative signs? Sample answer: The amount of each coupon is subtracted
____________ PERIOD _____
All real numbers can be classified as either rational or irrational. The
set of rational numbers includes several subsets: natural numbers, whole numbers,
and integers.
Properties of Real Numbers
Study Guide and Intervention
Real Numbers
1-2
NAME ______________________________________________ DATE
5/19/06
Read the introduction to Lesson 1-2 in your textbook.
____________ PERIOD _____
A2-01-873971
Get Ready for the Lesson
1-2
NAME ______________________________________________ DATE
Lesson 1-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A5
(Lesson 1-2)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Properties of Real Numbers
Study Guide and Intervention
A6
3
4
4a 3b
a
b
5. 12 3
4
10.2r 39.2s
6. 8(2.4r 3.1s) 6(1.5r 2.4s)
2
k
j
5
4.2k 5.4j
3
5
1
2
190a 70b
Chapter 1
23. 3(m z) 5(2m z) 13m 8z
21. (3g 3h) 5g 10h 2g 13h
19. 3x 5 2x 3 5x 2
Simplify each expression.
14
Glencoe Algebra 2
5
4
17. , 4 4
5 5
1
15
15. 15 15, Chapter 1
2j 7
Comm. ()
3 4
4 15
1
2
15
Glencoe Algebra 2
26. (15d 3) (8 10d) 10d 3
1
3
24. 2x 3y (5x 3y 2z) 3x 2z
22. a2 a 4a 3a2 1 2a2 3a 1
20. x y z y x z 0
3
4
18. 3 3 , 16. 1.25 1.25, 0.8
Name the additive inverse and multiplicative inverse for each number.
Assoc. ()
14. (10b 12b) 7b (12b 10b) 7b
Mult. Iden.
13. 0.6[25(0.5)] [0.6(25)]0.5
12. 15x(1) 15x
Mult. Inv.
Assoc. ()
10. 2r (3r 4r) (2r 3r) 4r
Add. Iden.
8. 3a 0 3a
11. 5y 1
5y1 Distributive
9. 2(r w) 2r 2w
25. 6(2 v) 4(2v 1) 8 2v
20 2n
2
16. (18 6n 12 3n)
3
40 105c
14. 2(15 45c) (12 18c)
5
6
1
4
p r
4
5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
18. 50(3a b) 20(b 2a)
8.75m
1
5
12. p r r p
3
4
12.7x 16
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
17. 14( j 2) 3j(4 7)
0.7x 9
15. (7 2.1x)3 2(3.5x 6)
140g 120h
13. 4(10g 80h) 20(10h 5g)
62.4e 39f
10. 9(7e 4f) 0.6(e 5f ) 11. 2.5m(12 8.5)
77 4p
7. 4(20 4p) (4 16p) 8. 5.5j 8.9k 4.7k 10.9j 9. 1.2(7x 5) (10 4.3x)
80g 26h
4. 10(6g 3h) 4(5g h)
51s 13t
2. 40s 18t 5t 11s
1
3. (4j 2k 6j 3k)
5
Commutative Property ()
Distributive Property
Simplify.
Comm. ()
7. 3 x x 3
Answers
20a
1. 8(3a b) 4(2b a)
Simplify each expression.
Exercises
9x 3y 12y 0.9x 9x ( 0.9x) 3y 12y
(9 ( 0.9))x (3 12)y
8.1x 15y
6. 30
I, R
5. 9
Z, Q, R
Name the property illustrated by each equation.
4. N, W, Z, Q, R
12
3
2. 525 Z, Q, R
3. 0.875 Q, R
1. 34 N, W, Z, Q, R
____________ PERIOD _____
8:03 AM
Simplify 9x 3y 12y 0.9x.
a(b c) ab ac and (b c)a ba ca
Distributive
Properties of Real Numbers
Skills Practice
Name the sets of numbers to which each number belongs.
1-2
NAME ______________________________________________ DATE
5/19/06
Example
a (a) 0 (a) a
Inverse
1
a
If a is not zero, then a 1 a.
a0a0a
Identity
1
a
(a b) c a (b c)
a
1a1
a
(a b) c a (b c)
Associative
Multiplication
a
bb
a
Addition
abba
Commutative
Property
For any real numbers a, b, and c
Real Number Properties
(continued)
____________ PERIOD _____
A2-01-873971
Properties of Real Numbers
1-2
NAME ______________________________________________ DATE
Lesson 1-2
A1-A23
Page A6
(Lesson 1-2)
Glencoe Algebra 2
Chapter 1
Properties of Real Numbers
Practice
25
36
Q, R
6. 16
Z, Q, R
I, R
2. 7
7. 35 Z, Q, R
I, R
3. 2
Distributive
17. 5(x y) 5x 5y
Comm. ()
14. 3x 2y 3 2 x y
Distributive
Add. Iden.
18. 4n 0 4n
Add. Inv.
15. (6 6)y 0y
12. 7n 2n (7 2)n
Assoc. ()
10. 7x (9x 8) (7x 9x) 8
A7
16
11
5 3
6 5
29. 2(4 2x y) 4(5 x y)
1
2
1
4
Chapter 1
16
1
1
false; counterexample: 5 4 5
4
a1 1b Glencoe Algebra 2
Chapter 1
Rationals
Irrationals
Real Numbers
3. VENN DIAGRAMS Make a Venn
diagram that shows the relationship
between natural numbers, integers,
rational numbers, irrational numbers,
and real numbers.
Integers
statement is true or false: If a b, it follows that a b . Explain your reasoning.
= 7
10
Commutative Property of
Multiplication
32. NUMBER THEORY Use the properties of real numbers to tell whether the following
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Natural
Numbers
13y
30. x 12y (2x 12y)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
31. TRAVEL Olivia drives her car at 60 miles per hour for t hours. Ian drives his car at
50 miles per hour for (t 2) hours. Write a simplified expression for the sum of the
distances traveled by the two cars. (110t 100) mi
12 8x 6y
1
5
28. (10a 15) (8 4a) 4a 1
27. 3(r 10s) 4(7s 2r) 5r 58s
26. 4c 2c (4c 2c) 4c
25. 8x 7y (3 6y) 8x y 3
5 6
5 ,
6 35
24. 11a 13b 7a 3b 4a 16b
5
22. 5 6
20. 1.6 1.6, 0.625
23. 5x 3y 2x 3y 3x
Simplify each expression.
11 11
21. ,
16 16
10
7
2. MODELS What property of real
numbers is illustrated by the figure
below?
Distributive Property
17
b
c
Answers
Glencoe Algebra 2
c 25; it is a natural number.
7. a 7, b 24
c 245
or 75
; it is not a
natural number.
6. a 7, b 14
c 13; it is a natural number.
5. a 5, b 12
For each set of values for a and b,
determine the value of c. State whether
c is a natural number.
a
The lengths of the sides of the right triangle
shown are related by the formula
c2 = a2 b2.
RIGHT TRIANGLES For Exercises 5–7,
use the following information.
I. always II. sometimes,
· 2
2
2
Answers
19. 0.4 0.4, 2.5
Name the additive inverse and multiplicative inverse for each number.
Mult. Inv.
16. 4y 1y
1
4
Assoc. ()
13. 3(2x)y (3 2)(xy)
Mult. Iden.
11. 5(3x y) 5(3x 1y)
Comm. ()
9. 5x (4y 3x) 5x (3x 4y)
8. 31.8 Q, R
W, Z, Q, R
4. 0
8:03 AM
Name the property illustrated by each equation.
5.
N, W, Z, Q, R
____________ PERIOD _____
4. NUMBER THEORY Consider the
following two statements.
I. The product of any two rational
numbers is always another rational
number.
II. The product of two irrational numbers
is always irrational.
Determine if these statements are
always, sometimes, or never true.
Explain.
Properties of Real Numbers
Word Problem Practice
1. MENTAL MATH When teaching
elementary students to multiply and
learn place value, books often show
that 54 8 (50 4) 8 (50 8) (4 8). What property is used?
1-2
NAME ______________________________________________ DATE
5/19/06
1. 6425
____________ PERIOD _____
A2-01-873971
Name the sets of numbers to which each number belongs.
1-2
NAME ______________________________________________ DATE
Lesson 1-2
A1-A23
Page A7
(Lesson 1-2)
Glencoe Algebra 2
Enrichment
Chapter 1
Does each number, a, have an inverse, a , such that
a a a a i? The integer inverse of 3 is 3 since
3 3 0, and 0 is the identity for addition. But the set does not
contain 3. Therefore, there is no inverse for 3.
Inverse:
A8
1 is the inverse of 1 since (1)(1) 1, and 1 is the identity.
1 is the inverse of 1 since (1)(1) 1, and 1 is the identity.
Each member has an inverse.
Inverse:
Chapter 1
Glencoe Algebra 2
8. {rational numbers}, addition yes
7. {irrational numbers}, addition no
18
6. {1
, 2
, 3
, …}, multiplication no
5. {x, x2, x3, x4, …} addition no
4. {multiples of 5}, multiplication no
2 3
2 2
3. , , , … , addition no
12
2. {integers}, multiplication no
1. {integers}, addition yes
Tell whether the set forms a group with respect to the given operation.
Exercises
The set {1, 1} is a group with respect to multiplication because all four properties hold.
1(1) 1; 1(1) 1
The identity for multiplication is 1.
Identity:
Associative Property: (1)[(1)(1)] (1)(1) 1; and so on
The set is associative for multiplication.
Is the set {1, 1} a group with respect to multiplication?
Closure Property:
(1)(1) 1; (1)(1) 1; (1)(1) 1; (1)(1) 1
The set has closure for multiplication.
Example 2
Chapter 1
19
Glencoe Algebra 2
Sample answer: When you look at your reflection, you are looking at
yourself. The reflexive property says that every number is equal to itself.
In geometry, symmetry with respect to a line means that the parts of a
figure on the two sides of a line are identical. The symmetric property of
equality allows you to interchange the two sides of an equation. The
equal sign is like the line of symmetry.
3. How can the words reflection and symmetry help you remember and distinguish between
the reflexive and symmetric properties of equality? Think about how these words are
used in everyday life or in geometry.
Remember What You Learned
2. When Louisa rented a moving truck, she agreed to pay $28 per day plus $0.42 per mile.
If she kept the truck for 3 days and the rental charges (without tax) were $153.72, how
many miles did Louisa drive the truck? 3(28) 0.42m 153.72
Read the following problem and then write an equation that you could use to
solve it. Do not actually solve the equation. In your equation, let m be the number
of miles driven.
Sample answer: An equation is a statement that says that two
algebraic expressions are equal.
c. How are algebraic expressions and equations related?
Sample answer: Equations contain equal signs; expressions do not.
b. How are algebraic expressions and equations different?
Sample answer: Both contain variables, constants, and operation
signs.
1. a. How are algebraic expressions and equations alike?
Read the Lesson
Answers
The set is not a group with respect to addition because only three of the four properties hold.
Is there some number, i, in the set such that i a a a i
for all a? 0 1 1 1 0; 0 2 2 2 0; and so on.
The identity for addition is 0.
(220 A) I
P or P (220 A) I 6
6
• Write an equation that shows how to calculate your target heart rate.
and desired intensity level (I )
• To find your target heart rate, what two pieces of information must you supply? age (A)
Read the introduction to Lesson 1-3 in your textbook.
____________ PERIOD _____
8:03 AM
Identity:
Associative Property: For all numbers in the set, does a (b c) (a b) c?
0 (1 2) (0 1) 2; 1 (2 3) (1 2) 3; and so on.
The set is associative for addition.
Does the set {0, 1, 2, 3, …} form a group with respect to addition?
Closure Property:
For all numbers in the set, is a b in the set? 0 1 1, and 1 is
in the set; 0 2 2, and 2 is in the set; and so on. The set has
closure for addition.
Solving Equations
Lesson Reading Guide
Get Ready for the Lesson
1-3
NAME ______________________________________________ DATE
5/19/06
Example 1
A set of numbers forms a group with respect to an operation if for that operation
the set has (1) the Closure Property, (2) the Associative Property, (3) a member
which is an identity, and (4) an inverse for each member of the set.
____________ PERIOD _____
A2-01-873971
Properties of a Group
1-2
NAME ______________________________________________ DATE
Lesson 1-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A8
(Lessons 1-2 and 1-3)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Solving Equations
Study Guide and Intervention
____________ PERIOD _____
division
times, product, of (as in of a number)
divided by, quotient
25
A9
n
Chapter 1
5n
11. n 8
n3
20
Glencoe Algebra 2
The quotient of five times a number and the sum of the
number and 3 is equal to the difference of the number and 8.
square of the number is equal to four times the number.
10. 2(n3 3n2) 4n Twice the sum of the cube of a number and three times the
9. 3n 35 79 The difference of three times a number and 35 is equal to 79.
Write a verbal sentence to represent each equation. Sample answers are given.
8. 23 more than the product of 7 and a number 7n 23
7. four times the square of a number increased by five times the same number 4n 2 5n
6. the product of 3 and the sum of 11 and a number 3(11 n)
5. the sum of 100 and four times a number 100 4n
4. the difference of nine times a number and the quotient of 6 and the same number
3. 7 less than fifteen times a number 15n 7
140
140 100
40
5
Solve 100 8x 140.
c
1 3
2 4
4
5
k
2
d
2
20. 2xy x 7, for x x Answers
Glencoe Algebra 2
21
10
3
Chapter 1
4
3
25. 4x 3y 10, for y y x 20n
5n 1
23. 3.5s 42 14t, for s s 4t 12
24. 5m 20, for m m m
n
22. 3(2j k) 108, for j j 18 f
4
4r
q
21. 6, for f f 24 2d
3pq
r
7
2y 1
s
20
19. 12, for p p s
2t
17. 10, for t t 18. h 12g 1, for g g h1
12
16. a 3b c, for b b ac
3
Solve each equation or formula for the specified variable.
15. 2x 75 102 x 9
14. 100 20 5r 16
13. 4n 20 53 2n 5 9. 5(4 k) 10k 4
6. 8 2(z 7) 3
3. 5t 1 6t 5 4
y 20 x
12. 4.5 2p 8.7 2.1
1
2
1
5
y (100 4x)
11. n 98 n 28
5
2
8. 3x 17 5x 13 15
5. 7 x 3 8
2. 17 9 a 8
1
2
Solve 4x 5y 100 for y.
10. 120 y 60 80
3
4
7. 0.2b 10 50
2
3
4. m 1. 3s 45 15
Example 2
4x 5y 100
4x 5y 4x 100 4x
5y 100 4x
Solve each equation. Check your solution.
Exercises
100 8x
100 8x 100
8x
x
Example 1
c
For any real numbers a, b, and c, if a b,
a
b
then a c b c and, if c is not zero, .
Multiplication and Division
Properties of Equality
Answers
2. four times the sum of a number and 3 4(n 3)
1. the sum of six times a number and 25 6n
6
9n Six times the difference of a number and two
is equal to 14.
Write an algebraic expression to represent each verbal expression.
Exercises
n
18
3
Example 1
Write an algebraic
expression to represent 18 less than
the quotient of a number and 3.
Example 2
Write a verbal sentence to
represent 6(n 2) 14.
multiplication
minus, difference, decreased by, less than
For any real numbers a, b, and c, if a b,
then a c b c and a c b c.
Addition and Subtraction
Properties of Equality
8:03 AM
1
2
addition
subtraction
and, plus, sum, increased by, more than
Operation
Word Expression
multiplication, or division.
(continued)
You can solve equations by using addition, subtraction,
Solving Equations
Study Guide and Intervention
____________ PERIOD _____
5/19/06
Properties of Equality
1-3
NAME ______________________________________________ DATE
A2-01-873971
Verbal Expressions to Algebraic Expressions The chart suggests some ways to
help you translate word expressions into algebraic expressions. Any letter can be used to
represent a number that is not known.
1-3
NAME ______________________________________________ DATE
Lesson 1-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A9
(Lesson 1-3)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Solving Equations
Skills Practice
____________ PERIOD _____
A10
y
3
A number divided by 3 is the
difference of 2 and twice the
number.
10. 2 2y
22. 2.2n 0.8n 5 4n 5
2a
21. a 3
5
Chapter 1
22
Glencoe Algebra 2
A 2
r 2
2
r
26. A 2r2 2rh, for h h xy
25. A , for y
2
y 2A x
1
24. y x 12, for x
4
I
23. I prt, for p p rt
Solve each equation or formula for the specified variable.
x 4y 48
20. 4v 20 6 34 5
19. 5x 3x 24 3
5
18. 3b 7 15 2b 17. 3t 2t 5 5
22
5
16. x 4 5x 2 15. 4m 2 18 4
1
2
Substitution ()
14. If (8 7)r 30, then 15r 30.
Subtraction ()
12. If d 1 f, then d f 1.
Solve each equation. Check your solution.
Symmetric ()
13. If 7x 14, then 14 7x.
Transitive ()
11. If a 0.5b, and 0.5b 10, then a 10.
are given.
The sum of 8 and 3 times a
number is 5.
8. 8 3x 5
are given.
of a number and 5 is 3 times the
sum of twice the number and 1.
8. 3(2m 1) The quotient
m
5
Three times a number is 4 times
the cube of the number.
6. 3y 4y3
1
2
5 1
8 4
1
6
3c 1
2
2
Chapter 1
23
Glencoe Algebra 2
26. GOLF Luis and three friends went golfing. Two of the friends rented clubs for $6 each. The
total cost of the rented clubs and the green fees for each person was $76. What was the cost
of the green fees for each person? g green fees per person; 6(2) 4g 76; $16
25. GEOMETRY The length of a rectangle is twice the width. Find the width if the
perimeter is 60 centimeters. w width; 2(2w) 2w 60; 10 cm
2(E U )
w
1
24. E Iw2 U, for I I 2
2d 1
3
22. c , for d d Define a variable, write an equation, and solve the problem.
E
c
h gt 2
23. h vt gt2, for v v t
21. E mc2, for m m 2
Solve each equation or formula for the specified variable.
20. 6y 5 3(2y 1) 4
5
19. 5(6 4v) v 21 3
7
11 1
12 5
18. 6x 5 7 9x 3
4
16. s 5
6
14. 9 4n 59 17
Multiplication ()
12. If 4m 15, then 12m 45.
Division ()
10. If 8(2q 1) 4, then 2(2q 1) 1.
17. 1.6r 5 7.8 8
15. n 3
4
13. 14 8 6r 1
Solve each equation. Check your solution.
Subtraction ()
11. If h 12 22, then h 10.
Symmetric ()
9. If t 13 52, then 52 t 13.
Name the property illustrated by each statement.
Three times a number is twice the
difference of the number and 1.
7. 3c 2(c 1)
The difference of 5 and twice a
number is 4.
5. 5 2x 4
Answers
Name the property illustrated by each statement.
Three added to the square of
a number is the number.
9. b2 3 b
The difference of a number
and 8 is 16.
7. n 8 16
2y 2 1
4. 1 less than twice the square of a number
n (n 1)
2. the sum of two consecutive integers
Write a verbal expression to represent each equation. 5–8. Sample answers
5(m 1)
3. 5 times the sum of a number and 1
y
2
5
1. 2 more than the quotient of a number and 5
Write an algebraic expression to represent each verbal expression.
8:03 AM
Write a verbal expression to represent each equation. 7–10. Sample answers
6. the product of 11 and the square of a number 11n2
5. 3 times the difference of 4 and a number 3(4 n)
3n
9
4. the product of 3 and a number, divided by 9
5n 8
2. 8 less than 5 times a number
Solving Equations
Practice
____________ PERIOD _____
5/19/06
6(n 5)
3. 6 times the sum of a number and 5
4n 7
1. 4 times a number, increased by 7
1-3
NAME ______________________________________________ DATE
A2-01-873971
Write an algebraic expression to represent each verbal expression.
1-3
NAME ______________________________________________ DATE
Lesson 1-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A10
(Lesson 1-3)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Solving Equations
A11
Chapter 1
w 6 units
w
w 10
3. GEOMETRY The length of a rectangle
is 10 units longer than its width. If
the total perimeter of the rectangle is
44 units, what is the width?
24
3t
80 dominoes
Glencoe Algebra 2
6. How many dominoes did Nancy have in
her hallway?
321 4N 1.
5. Nancy measures her dominoes and finds
that t 1 centimeter. She measures the
distance of her hallway and finds that
d 321 centimeters. Rewrite the
equation that relates d, t, and N with the
given values substituted for t and d.
t
Chapter 1
25
Answers
Glencoe Algebra 2
There is a surplus. Consumption is reduced, possibly because the cost
of living has reduced.
4. Determine if there is a trade surplus or deficit when there is 12 trillion dollar GNP,
2 trillion in investments, 3 trillion in government investments, and 5 trillion in
consumption. Explain why this situation may be favorable.
Consumption will increase.
3. If the GNP remains steady, and so do investments and government spending, but the
trade deficit increases (to say 2 or 3 trillion dollars), what does this say about the
consumption level?
Government spending must have gone up, in fact G 6 trillion dollars.
Employment may have caused a dip in consumption.
2. In 2001, the U.S. trade deficit remain at 1 trillion dollars, investments also remain
steady at 1 trillion dollars. However consumption dipped to only 50% of the GNP,
which increased to 12 trillion dollars. What was the effect on government spending?
What might have caused the change?
10.5 6.3 1 G 1. Therefore G 4.2 trillion dollars.
1. The most important sector of the U.S. economy is consumption. It makes up about
60% of the entire GNP. In 2000, the U.S.’s GNP was 10.5 trillion dollars. In the same
year, there were 1 trillion dollars in investments, but a 1 trillion dollar trade deficit.
Assuming that consumption made up 60% of the GNP, how much did the government
budget for spending?
Answers
20 suitcases
2. AIRPLANES The number of passengers
p and the number of suitcases s that an
airplane can carry are related by the
equation 180p + 60s = 3,000. If 10
people board the aircraft, how many
suitcases can the airplane carry?
Nancy is setting up a train of dominos from
the front entrance straight down the hall to
the kitchen entrance. The thickness of each
domino is t. Nancy places the dominoes so
that that the space separating consecutive
dominoes is 3t. The total distance that N
dominoes takes up is given by d = t(4N + 1).
the information below.
C is consumer goods (e.g. TV’s, Cars, Food, Furniture, Clothes, Doctors’ fees, and
Dining)
I is investments (e.g. Factories, Computers, Airlines, and Housing)
G is government spending and investments (e.g. Ships, Roads, Schools, NASA,
and Bombs)
X is exports (e.g. Corn, Wheat, Cars, and Computers)
M is for imports, (e.g. Cars, Computer chips, Clothes, and Oil)
Calculated from GNP C I G X M, where
8:03 AM
DOMINOES For Exercises 5 and 6, use
15
United States’ Gross National Product
Enrichment
____________ PERIOD _____
The Gross National Product, GNP, is an important indicator of U.S. economy. The
GNP contains information about the inflation rate, the Bond market, and the Stock
market. It is composed of consumer goods, investments, government expenditures,
exports, and imports.
1-3
NAME ______________________________________________ DATE
5/19/06
F 3R 5.
4. SAVINGS Jason started with d dollars in
his piggy bank. One week later, Jason
doubled the amount in his piggy bank.
Another week later, Jason was able to add
$20 to his piggy bank. At this point, the
piggy bank had $50 in it. What is d?
Word Problem Practice
____________ PERIOD _____
A2-01-873971
1. AGES Robert’s father is 5 years older
than 3 times Robert’s age. Let Robert’s
age be denoted by R and let Robert’s
father’s age be denoted by F. Write an
equation that relates Robert’s age and
his father’s age.
1-3
NAME ______________________________________________ DATE
Lesson 1-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A11
(Lesson 1-3)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Solving Equations and Checking Solutions
Graphing Calculator Activity
____________ PERIOD _____
Solve 2(5y 1) y 4(y 3).
ENTER
3
ENTER
—
)
2
5
2nd
—
1
6 ZOOM
[QUIT]
ZOOM
(
8
)
MATH
ENTER
—
2nd
ENTER
(–)
[CALC] 5
.
ENTER
ENTER
4
2
A12
ZOOM
ENTER
VARS
2nd
—
6
)
(
[CALC] 2.
)
2
—
ENTER
5
—
2
)
(
ENTER
4
ENTER
VARS
)
11
Chapter 1
4
z 1
4. 3(2z 25) 2(z 1) 78
2
w 5
1. 3(2w 7) 9 2(5w 4)
Solve each equation.
Exercises
26
m 8
3
5
m4
3m 1
5. 1
x 17
2. 1.5(4 x) 1.3(2 x)
2
2
8
11
21
x
Glencoe Algebra 2
x5
1
x3
6. 2x 5
a 4
4
3. 1(a 2) 1(5 a)
6
[10, 10] scl:1 by [10, 10] scl:1
20
Use arrow keys and enter to set the bound prompts. The solution is x .
ENTER
ENTER
Keystrokes: Y = (
( 1 2
ENTER
Graph the expression on the left side of the equation in Y1 and the
expression on the right side of the equation in Y2. Enter Y1 - Y2 in
Y3. Then graph the function in Y3. Use the zero function under the
CALC menu to determine where the graph of Y3 equals zero. This
point will be the solution.
5
Solve x x 1(x 2).
[47, 47] scl:10 by [31, 31] scl:10
Chapter 1
27
Glencoe Algebra 2
Sample answer: The number line shows that for every positive number,
there are two numbers that have that number as their absolute value.
5. How can the number line model for absolute value that is shown on page 28 of your
textbook help you remember that many absolute value equations have two solutions?
Remember What You Learned
is , then there are no solutions.
4. What does the symbol mean as a solution set? Sample answer: If a solution set
greater than 0.
3. What does the sentence b 0 mean? Sample answer: The number b is 0 or
The absolute value is the number of units it is from 0 on the number line.
The number of units is never negative.
2. Explain why the absolute value of a number can never be negative. Sample answer:
answer: If a is negative, then a is positive. Example: If a 25, then
a (25) 25.
1. Explain how a could represent a positive number. Give an example. Sample
Read the Lesson
• If the magnitude of an earthquake is estimated to be 6.9 on the Richter scale, it might
6.6 or as high as 7.2 .
actually have a magnitude as low as
from the measured magnitude by up to 0.3 unit in either direction, so an
absolute value equation is needed.
Answers
Example 2
7
10
is the value of both sides of the equation when x .
7
10
The x-coordinate, , is the solution to the equation. The y-coordinate
ENTER
(–)
2:55 PM
Y=
• Why is an absolute value equation rather than an equation without absolute value used
to find the extremes in the actual magnitude of an earthquake in relation to its measured
value on the Richter scale? Sample answer: The actual magnitude can vary
who studies earthquakes; a number from 1 to 10 that tells how strong an
earthquake is
• What is a seismologist and what does magnitude of an earthquake mean? a scientist
Read the introduction to Lesson 1-4 in your textbook.
5/25/06
Keystrokes:
4 (
Graph the expression on the left side of the equation in Y1 and
the expression on the right side of the equation in Y2. Choose an
appropriate view window so that the intersection of the graphs is
visible. Then use the intersect command to find the coordinates of
the common point.
Example 1
Solving Absolute Value Equations
Lesson Reading Guide
Get Ready for the Lesson
1-4
NAME ______________________________________________ DATE______________ PERIOD _____
A2-01-873971
When solving equations, checking the solutions is an important process. A
graphing calculator can be used to check the solution of an equation.
1-3
NAME ______________________________________________ DATE
Lesson 1-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A12
(Lessons 1-3 and 1-4)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Solving Absolute Value Equations
Study Guide and Intervention
1
2
⏐2x 3y⏐ ⏐2(4) 3(3)⏐
⏐8 9⏐
⏐17⏐
17
Evaluate 2x 3y if
x 4 and y 3.
Example 2
A13
Chapter 1
22. ⏐yz 4w⏐ w 17
1
2
28
23. ⏐wz⏐ ⏐8y⏐ 20
3
4
20. 12 ⏐10x 10y⏐ 3
17. ⏐2x y⏐ 5y 6
16. 14 2⏐w xy⏐ 4
19. z⏐z⏐ x⏐x⏐ 32
14. 3⏐wx⏐ ⏐4x 8y⏐ 27
13. ⏐6y z⏐ ⏐yz⏐ 6
1
4
11. ⏐z⏐ 4⏐2z y⏐ 40
8. ⏐wz⏐ ⏐xy⏐ 23
7. ⏐w 4x⏐ 12
10. 5⏐w⏐ 2⏐z 2y⏐ 34
5. ⏐x⏐ ⏐y⏐ ⏐z⏐ 4 4. ⏐x 5⏐ ⏐2w⏐ 1
1
2
2. ⏐6 z⏐ ⏐7⏐ 7
Glencoe Algebra 2
24. xz ⏐xz⏐ 24
1
2
21. ⏐5z 8w⏐ 31
18. ⏐xyz⏐ ⏐wxz⏐ 54
15. 7⏐yz⏐ 30 9
12. 10 ⏐xw⏐ 2
9. ⏐z⏐ 3⏐5yz⏐ 39
6. ⏐7 x⏐ ⏐3x⏐ 11
3. 5 ⏐w z⏐ 15
a
2x 3
2x 3 3
2x
x
b
17
17 3
20
10
1
7
b
17
17 3
14
7
11
, 1
2
Chapter 1
1
2
12 15. ⏐6 2x⏐ 3x 1 29
Answers
16. ⏐16 3x⏐ 4x 12 {4}
Glencoe Algebra 2
14. ⏐4b 3⏐ 15 2b {2, 9}
13. 5f ⏐3f 4⏐ 20 {12}
10. ⏐3x 1⏐ 2x 11 {2, 12}
8. ⏐8 5a⏐ 14 a
6. ⏐15 2k⏐ 45 {15, 30}
4. ⏐m 3⏐ 12 2m {3}
2. ⏐t 4⏐ 5 0 {1, 9}
⏐14 3⏐ 17
⏐17⏐ 17
17 17 ✓
12. 40 4x 2⏐3x 10⏐ {6, 10}
⏐
11. x 3 1 1
⏐3
9. ⏐4p 11⏐ p 4 23, 1
3
7. 5n 24 ⏐8 3n⏐ {2}
5. ⏐5b 9⏐ 16 2 3. ⏐x 5⏐ 45 {40, 50}
1. ⏐x 15⏐ 37 {52, 22}
Solve each equation. Check your solutions.
Exercises
a
2x 3
2x 3 3
2x
x
CHECK ⏐2(7) 3⏐ 17
Case 2
Solve 2x 3 17. Check your solutions.
⏐2x 3⏐ 17
⏐2(10) 3⏐ 17
⏐20 3⏐ 17
⏐17⏐ 17
17 17 ✓
There are two solutions, 10 and 7.
CHECK
Case 1
Example
Answers
1. ⏐2x 8⏐ 4
Evaluate each expression if w 4, x 2, y , and z 6.
Exercises
⏐4⏐ ⏐2x⏐ ⏐4⏐ ⏐2 6⏐
⏐4⏐ ⏐12⏐
4 12
8
x 6.
Evaluate 4 2x if
For any real numbers a and b, where b 0, if ⏐a⏐ b then a b or a b.
Always check your answers by substituting them into the original equation. Sometimes
computed solutions are not actual solutions.
8:03 AM
Example 1
Absolute Value
For any real number a, if a is positive or zero, the absolute value of a is a.
If a is negative, the absolute value of a is the opposite of a.
• Symbols For any real number a, ⏐a⏐ a, if a 0, and ⏐a⏐ a, if a 0.
• Words
(continued)
Use the definition of absolute value to solve equations
containing absolute value expressions.
Absolute Value Equations
Solving Absolute Value Equations
Study Guide and Intervention
5/19/06
The absolute value of a number is the number of
units it is from 0 on a number line. The symbol ⏐x⏐ is used to represent the absolute value
of a number x.
1-4
NAME ______________________________________________ DATE______________ PERIOD _____
A2-01-873971
Absolute Value Expressions
1-4
NAME ______________________________________________ DATE______________ PERIOD _____
Lesson 1-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A13
(Lesson 1-4)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Solving Absolute Value Equations
Skills Practice
____________ PERIOD _____
6. ⏐8x 3y⏐ ⏐2y 5x⏐ 21
8. 44 ⏐2x y⏐ 45
5. ⏐10z 31⏐ 131
7. 25 ⏐5z 1⏐ 24
12. 6.4 ⏐w 1⏐ 7
A14
1 5
2 6
5 1
6 6
Chapter 1
25. 5⏐6a 2⏐ 15 , 1
2
30
26. ⏐k⏐ 10 9 Glencoe Algebra 2
24. ⏐8d 4d⏐ 5 13 {2, 2}
22. 4⏐7 y⏐ 1 11 {4, 10}
21. ⏐7x 3x⏐ 2 18 {4, 4}
23. ⏐3n 2⏐ , 20. 2⏐3w⏐ 12 {2, 2}
19. ⏐p 7⏐ 14 16. ⏐2g 6⏐ 0 {3}
18. ⏐2x x⏐ 9 {3, 3}
17. 10 ⏐1 c⏐ {9, 11}
4 8
15. ⏐3k 6⏐ 2 , 3 3
14. ⏐5a⏐ 10 {2, 2}
12. ⏐2 2d⏐ 3⏐b⏐ 19.2
16. 7⏐x 3⏐ 42 {9, 3}
18. ⏐5x 4⏐ 6 15. ⏐2y 3⏐ 29 {13, 16}
17. ⏐3u 6⏐ 42 {12, 16}
2
3
28. 3 5⏐2d 3⏐ 4 27. 5⏐2r 3⏐ 5 0 {2, 1}
Chapter 1
x 51 3 or 48 x 54
31
Glencoe Algebra 2
30. OPINION POLLS Public opinion polls reported in newspapers are usually given with a
margin of error. For example, a poll with a margin of error of 5% is considered accurate
to within plus or minus 5% of the actual value. A poll with a stated margin of error of
3% predicts that candidate Tonwe will receive 51% of an upcoming vote. Write and
solve an equation describing the minimum and maximum percent of the vote that
candidate Tonwe is expected to receive.
x 87.4 1.5; or 85.9 x 88.9
29. WEATHER A thermometer comes with a guarantee that the stated temperature differs
from the actual temperature by no more than 1.5 degrees Fahrenheit. Write and solve an
equation to find the minimum and maximum actual temperatures when the
thermometer states that the temperature is 87.4 degrees Fahrenheit.
26. 5 3⏐2 2w⏐ 7 {3, 1}
25. 2⏐4 s⏐ 3s {8}
24. 2⏐7 3y⏐ 6 14 1, 3 22. ⏐4w 1⏐ 5w 37 {38}
21. ⏐8 p⏐ 2p 3 {11}
23. 4⏐2y 7⏐ 5 9 {3, 4}
20. 6⏐5 2y⏐ 9 1.75, 3.25
19. 3⏐4x 9⏐ 24 14. ⏐x 13⏐ 2 {11, 15}
13. ⏐n 4⏐ 13 {9, 17}
Solve each equation. Check your solutions.
11. ⏐a b⏐ ⏐b a⏐ 14
Answers
13. ⏐y 3⏐ 2 {5, 1}
Solve each equation. Check your solutions.
11. ⏐3x 2y⏐ 4 4
8. ⏐1 7c⏐ ⏐a⏐ 33
7. ⏐5a 7⏐ ⏐3c 4⏐ 23
10. ⏐4d⏐ ⏐5 2a⏐ 12.6
6. ⏐2b 1⏐ ⏐8b 5⏐ 52
5. 6⏐10a 12⏐ 132
9. 3⏐0.5c 2⏐ ⏐0.5b⏐ 17.5
4. ⏐17c⏐ ⏐3b 5⏐ 114
3. ⏐10d a⏐ 15
8:03 AM
10. 3 ⏐1 6w⏐ 1.6
4. ⏐17z⏐ 170
3. ⏐9y z⏐ 17
2. ⏐2b 4⏐ 12
1. ⏐6a⏐ 6
5/19/06
9. 2⏐4w⏐ 3.2
2. ⏐9y⏐ 27
1. ⏐5w⏐ 2
Solving Absolute Value Equations
Practice
Evaluate each expression if a 1, b 8, c 5, and d 1.4.
1-4
NAME ______________________________________________ DATE______________ PERIOD _____
A2-01-873971
Evaluate each expression if w 0.4, x 2, y 3, and z 10.
1-4
NAME ______________________________________________ DATE
Lesson 1-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A14
(Lesson 1-4)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
12
14
16
18
20
22
24
A15
Chapter 1
|a 200| 3
a 197 or 203
32
Glencoe Algebra 2
They will be equal only if Jim walks
in the same direction each time
giving a, b, and c all the same sign.
7. When would the formula you wrote in
part A give the same value as the formula
shown in part B?
The distance Jim ends up from
where he started.
6. The equation you wrote in part A
should not be T |a b c|.
What does |a b c| represent?
T |a| |b | |c |
5. Write a formula for the total distance
that Jim walked.
Jim is walking along a straight line. An
observer watches him. If Jim walks forward,
the observer records the distance as a
positive number, but if he walks backward,
the observer records the distance as a
negative number. The observer has recorded
that Jim has walked a, then b, then c feet.
WALKING For Exercises 5–7, use the
following information.
x 2 3 (x 6)
(x 2) 3 x 6
4. | x 4 | 6 | x 3 | x 2.5
3. |3x 6 | |5x 10 | x 2
Chapter 1
Answers
(x 2) (2x 4) (x 3)
Glencoe Algebra 2
x 2 (2x 4) x 3
x 2 (2x 4) (x 3)
(x 2) (2x 4) x 3
33
(x 2) 2x 4 (x 3)
x 2 2x 4 (x 3)
no solution
(x 2) 2x 4 x 3
x 2 2x 4 x 3
6. List each case and solve | x 2 | | 2x 4 | | x 3 |. Check your solution.
5. How many cases would there be for an absolute value equation containing
three sets of absolute value symbols? 8
2. |2x 9 | | x 3 | x 12, 2
1. | x 4 | | x 7 | x 1.5
Solve each absolute value equation. Check your solution.
Exercises
| x 2 | 3 | x 6 |. The only solution to this equation is 52.
Solve each of these equations and check your solutions in the original equation,
x 2 3 (x 6)
x23x6
Each of these equations also has two cases. By writing the equations for both
cases of each equation above, you end up with the following four equations:
Answers
3. AGES Rhonda conducts a survey of the
ages of students in eleventh grade at her
school. On November 1, she finds the
average age is 200 months. She also
finds that two-thirds of the students
are within 3 months of the average
age. Write and solve an equation to
determine the age limits for this group
of students.
d |s j| or d |j s|
2. HEIGHT Sarah and Jessica are sisters.
Sarah’s height is s inches and Jessica’s
height is j inches. Their father wants
to know how many inches separate
the two. Write an equation for this
difference in such a way that the result
will always be positive no matter which
sister is taller.
13 or 21
10
| x 2| 3 x 6
| x 2 | 3 (x 6)
Consider the problem | x 2 | 3 | x 6 |. First we must write the equations
for the case where x 6 0 and where x 6 0. Here are the equations for
these two cases:
You have learned that absolute value equations with one set of absolute value
symbols have two cases that must be considered. For example, | x 3 | 5 must
be broken into x 3 5 or (x 3) 5. For an equation with two sets of
absolute value symbols, four cases must be considered.
8:03 AM
Maria’s
| w 10| 0.1
minimum weight: 9.9 pounds
maximum weight: 10.1 pounds
Enrichment
____________ PERIOD _____
Considering All Cases in Absolute Value Equations
1-4
NAME ______________________________________________ DATE
5/19/06
4. TOLERANCE Martin makes exercise
weights. For his 10 pound dumbbells, he
guarantees that the actual weight of his
dumbbells is within 0.1 pounds of 10
pounds. Write and solve an equation that
describes the minimum and maximum
weight of his 10 pound dumbbells.
Solving Absolute Value Equations
Word Problem Practice
____________ PERIOD _____
A2-01-873971
1. LOCATIONS Identical vacation
cottages, equally spaced along a street,
are numbered consecutively beginning
with 10. Maria lives in cottage #17.
Joshua lives 4 cottages away from
Maria. If n represents Joshua’s cottage
number, then |n 17| 4. What are
the possible numbers of Joshua’s
cottage?
1-4
NAME ______________________________________________ DATE
Lesson 1-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A15
(Lesson 1-4)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Absolute Value Statements
Spreadsheet Activity
A16
Chapter 1
34
5. If a and b are real numbers, then c|a b| c|a| |b|. sometimes
4. If a and b are real numbers, then |a| |b| a b. sometimes
3. If a and b are real numbers, then |a b| x. never
2. If a and b are real numbers, then |a b| |a| |b|. sometimes
1. For all real numbers a and b, a 0, |ax b| 0. sometimes
Use a spreadsheet to determine whether each absolute value
statement is sometimes, always, or never true.
Exercises
Through observation of Column D, when c is negative the statement is not
true. The absolute value statement, c|a b| |ca cb| is sometimes true;
it is true only if c 0.
Plan 2:
$55
2
3
4
5 4 3 2 1 0
5 4 3 2 1 0
⫺1 ⫺0 1
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
1
1
5
1
2
2
6
2
3
3
7
3
4
4
8
4
5
5
9
5
{x x 1}
{x x 2}
{x x 5}
{x x, 3}
Chapter 1
35
Glencoe Algebra 2
numbers, for example, 3 and 8. Then plot their additive inverses, 3 and
8. Write an inequality that compares the positive numbers and one that
compares the negative numbers. Notice that 8 3, but 8 3. The
order changes when you multiply by 1.
3. One way to remember something is to explain it to another person. A common student
error in solving inequalities is forgetting to reverse the inequality symbol when
multiplying or dividing both sides of an inequality by a negative number. Suppose that
your classmate is having trouble remembering this rule. How could you explain this rule
to your classmate? Sample answer: Draw a number line. Plot two positive
Remember What You Learned
d. There is space for at most 165 students in the high school band. 165
c. To participate in a concert, you must be willing to attend at least ten rehearsals. 10
b. A student may enroll in no more than six courses each semester. 6
a. There are fewer than 600 students in the senior class. 600
2. Show how you can write an inequality symbol followed by a number to describe each of
the following situations.
d.
c.
b.
a.
1. There are several different ways to write or show inequalities. Write each of the
following in interval notation.
Read the Lesson
Plan 2
Answers
Step 2 Use Column D to test
the equation. A formula
such as C2*ABS(A2B2)
ABS(C2*A2C2*B2)
in cell D2 returns TRUE
if the equation is true.
$65
Which plan should you choose?
Plan 1:
• Suppose that in one month you use 475 minutes of airtime on your wireless phone. Find
your monthly cost with each plan.
• Write an inequality comparing the number of minutes per month included in the two
phone plans. 150 400 or 400 150
Read the introduction to Lesson 1-5 in your textbook.
8:03 AM
Step 1 Use Columns A, B, and C
for the values of a, b, and
c. Choose several sets of
values including positive
and negative numbers,
and zero.
Solving Inequalities
Lesson Reading Guide
Get Ready for the Lesson
1-5
NAME ______________________________________________ DATE______________ PERIOD _____
5/19/06
Determine whether c|a b| |ca cb| is sometimes, always, or
never true.
Try a number of values for a, b, and c to determine whether the statement is
true or false for each set of values.
Glencoe Algebra 2
____________ PERIOD _____
A2-01-873971
You can use a spreadsheet to try several different values in an equation to
help you determine whether the statement is sometimes, always, or never
true. Remember that showing that a statement is true for some values does
not prove that it is true for all values. However, finding one value for which a
statement is false proves that it is not true for all values.
1-4
NAME ______________________________________________ DATE
Lesson 1-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A16
(Lessons 1-4 and 1-5)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
A17
1
2
Chapter 1
4 3 2 1 0
1
xx
2
1
2
3
7. 4x 2 7(4x 2)
3
4
4
8 7 6 5 4 3 2 1 0
{k k 4}
4. 18 4k 2(k 21)
4 3 2 1 0
{a a 3}
1. 7(7a 9) 84
9 8 7 6 5 4 3 2 1
17 3w 35
17 3w 17 35 17
3w 18
w 6
The solution set is {w⏐w 6}.
1
1
3
8
12
36
10
{y y 9}
14
3
4
8
6
9 10 11 12 13 14
2
8. (2y 3) y 2
7
{b b 11}
5. 4(b 7) 6 22
4 3 2 1 0
{z z 2}
2. 3(9z 4) 35z 4
6
c
c
c
Solve 17 3w 35. Then
graph the solution set on a number line.
Example 2
1
2
3
4
5
6
7
8
Glencoe Algebra 2
19 20 21 22 23 24 25 26 27
{d d 24}
9. 2.5d 15 75
0
{m m 5}
6. 2 3(m 5) 4(m 3)
8 7 6 5 4 3 2 1 0
{n n 7}
3. 5(12 3n) 165
Solve each inequality. Describe the solution set using set-builder notation.
Then graph the solution set on a number line.
Exercises
c
a
b
4. If c is negative and a b, then ac bc and .
is at least
is no less than
is greater than or equal to
Chapter 1
4 strings.
37
Answers
Glencoe Algebra 2
5. Solve the inequality and interpret the solution. The friends can bowl at most
4. Write an equation to describe this situation. 3(3.50) 4(1.50)n 40
Four friends plan to spend Friday evening at the bowling alley. Three of the friends need to
rent shoes for $3.50 per person. A string (game) of bowling costs $1.50 per person. If the
friends pool their $40, how many strings can they afford to bowl?
BOWLING For Exercises 4 and 5, use the following information.
per hour in order to finish the book in less than 6 hours.
3. Solve the inequality and interpret the solution. Ethan must read at least 67 pages
2. Write an inequality to describe this situation. 6 or 6n 482 80
482 80
n
Ethan is reading a 482-page book for a book report due on Monday. He has already read
80 pages. He wants to figure out how many pages per hour he needs to read in order to
finish the book in less than 6 hours.
PLANNING For Exercises 2 and 3, use the following information.
At most 17 hours
1. PARKING FEES The city parking lot charges $2.50 for the first hour and $0.25 for each
additional hour. If the most you want to pay for parking is $6.50, solve the inequality
2.50 0.25(x 1) 6.50 to determine for how many hours you can park your car.
Exercises
Let x be the number of remaining games that the Vikings must win. The total number of
games they will have won by the end of the season is 16 x. They want to win at least 80%
of their games. Write an inequality with .
16 x 0.8(36)
x 0.8(36) 16
x 12.8
Since they cannot win a fractional part of a game, the Vikings must win at least 13 of the
games remaining.
Example
SPORTS The Vikings play 36 games this year. At midseason, they
have won 16 games. How many of the remaining games must they win in order to
win at least 80% of all their games this season?
is at most
is no more than
is less than or equal to
Answers
13 14 15 16 17 18 19 20 21
2x 4 4 36 4
2x 32
x 16
The solution set is {x⏐x 16}.
Solve 2x 4 36.
Then graph the solution set on a
number line.
Example 1
These properties are also true for and .
c
a
b
3. If c is negative and a b, then ac bc and .
is greater than
is more than
is less than
is fewer than
8:03 AM
c
a
b
2. If c is positive and a b, then ac bc and .
c
For any real numbers a, b, and c, with c 0:
a
b
1. If c is positive and a b, then ac bc and .
For any real numbers a, b, and c:
1. If a b, then a c b c and a c b c.
2. If a b, then a c b c and a c b c.
c
Multiplication and Division Properties for Inequalities
Addition and Subtraction Properties for Inequalities
(continued)
Many real-world problems involve
inequalities. The chart below shows some common phrases that indicate inequalities.
Solving Inequalities
Study Guide and Intervention
Real-World Problems with Inequalities
1-5
5/19/06
The following properties can be used to solve inequalities.
Solving Inequalities
Study Guide and Intervention
NAME ______________________________________________ DATE______________ PERIOD _____
A2-01-873971
Solve Inequalities
1-5
NAME ______________________________________________ DATE______________ PERIOD _____
Lesson 1-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A17
(Lesson 1-5)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Solving Inequalities
Skills Practice
z
4
1
2
3
4
5
6
7
1
2
3
4
1
2
3
4
5
6
A18
1
2
3
4
1
2
3
4
5
6
1
2
3
4
5
6
4
1
2
3
4
1
2
3
4
5
6
1
2
3
4
1
2
3
4
3
4
1
2
3
4
1 0
1
2
3
4
5
6
7
14. 4x 9 2x 1 {x x 5}
4 3 2 1 0
12. 4(5x 7) 13 x x or
4 3 2 1 0
7
10. 7t (t 4) 25 t t 2
4 3 2 1 0
8. 7f 9 3f 1 {f f 2}
2 1 0
6. 4b 9 7 {b b 4}
4 3 2 1 0
Chapter 1
6n 5 2n; n 5
4
38
Glencoe Algebra 2
18. Five less than the product of 6 and a number is no more than twice that same number.
1
2
17. One half of a number is more than 6 less than the same number. n n 6; n 12
16. The difference of three times a number and 16 is at least 8. 3n 16 8; n 8
15. Nineteen more than a number is less than 42. n 19 42; n 23
Define a variable and write an inequality for each problem. Then solve.
2 1 0
13. 1.7y 0.78 5 {y y 3.4}
2 1 0
11. 0.7m 0.3m 2m 4 {m m 4}
4 3 2 1 0
3
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
5
6
1
2
3
4
1
2
3
4
1
2
3
4
5
6
1
3
4
5
4
1
2
3
4
1
2
3
4
1
2
3
4
5
7
8
4
3
2 1 0
1
{n n 4}
1
2
3
4
2
3
4
5
6
14. 4n 5(n 3) 3(n 1) 4
4 3 2 1 0
6
12. q 2(2 q) 0 q q 0
Chapter 1
39
Glencoe Algebra 2
18. BANKING Jan’s account balance is $3800. Of this, $750 is for rent. Jan wants to keep a
balance of at least $500. Write and solve an inequality describing how much she can
withdraw and still meet these conditions. 3800 750 w 500; w $2550
17. HOTELS The Lincoln’s hotel room costs $90 a night. An additional 10% tax is added.
Hotel parking is $12 per day. The Lincoln’s expect to spend $30 in tips during their stay.
Solve the inequality 90x 90(0.1)x 12x 30 600 to find how many nights the
Lincoln’s can stay at the hotel without exceeding total hotel costs of $600. 5 nights
4[2n (3)] 5.5n; n 4.8
16. Four times the sum of twice a number and 3 is less than 5.5 times that same number.
15. Twenty less than a number is more than twice the same number.
n 20 2n; n 20
10. 1 5(x 8) 2 (x 5) {x x 6}
4 3 2 1 0
8. 17.5 19 2.5x {x x 0.6}
4 3 2 1 0
2
6. 3(4w 1) 18 w w 4 3 2 1 0
4. 14s 9s 5 {s s 1}
2 1 0
Define a variable and write an inequality for each problem. Then solve.
4 3 2 1 0
{w w 3}
13. 36 2(w 77) 4(2w 52)
4 3 2 1 0
11. 3.5 {x x 1}
2
4x 3
2 1 0
9. 9(2r 5) 3 7r 4 {r r 4}
4 3 2 1 0
7. 1 8u 3u 10 {u u 1}
4 3 2 1 0
2
5. 9x 11 6x 9 x x 3
4 3 2 1 0
3. 16 8r 0 {r r 2}
4 3 2 1 0
2. 23 4u 11 {u u 3}
Answers
9. 3s 8 5s {s s 1}
2 1 0
7. 2z 9 5z {z z 3}
4 3 2 1 0
2
4. 20 3s 7s {s s 2}
1
1. 8x 6 10 {x x 2}
8:03 AM
5. 3x 9 {x x 3}
1 0
3. 16 3q 4 {q q 4}
4 3 2 1 0
2. 3a 7 16 {a a 3}
5/19/06
9 8 7 6 5 4 3 2 1
1. 2 {z z 8}
Solving Inequalities
Practice
Solve each inequality. Describe the solution set using set-builder or interval
notation. Then, graph the solution set on a number line.
1-5
NAME ______________________________________________ DATE______________ PERIOD _____
A2-01-873971
Solve each inequality. Describe the solution set using set-builder or interval
notation. Then, graph the solution set on a number line.
1-5
NAME ______________________________________________ DATE______________ PERIOD _____
Lesson 1-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A18
(Lesson 1-5)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Solving Inequalities
A19
Chapter 1
15P 300 1500
P 80; Manuel must translate at
least 80 pages.
40
125 pounds
Glencoe Algebra 2
7. Ron and his father want to go on the ride
together. Ron’s father weighs 175 pounds.
What is the maximum weight Ron can be
for the two to be allowed on the ride?
a 150
6. Write an inequality that describes the
limit on the average weight a of the two
riders.
x y 300
5. Write an inequality that describes the
weight limitation in terms of x and y.
On a Ferris wheel at a carnival, only two
people per car are allowed. The two people
together cannot weigh more than 300
pounds. Let x and y be the weights of the
people.
CARNIVALS For Exercises 5–7, use the
following information.
It is incorrect. From step 2 to step
3, Brandon must change
the direction of the inequality
because he is dividing by a
negative number.
The correct answer is x 6.
For all elements a, b, and c of set A,
if a R b and b R c, then a R c.
Transitive Property
, {all lines in a plane}
no; reflexive
Chapter 1
41
Answers
14. is the greatest integer not greater than, {all integers} yes
no; reflexive, symmetric, transitive
13. is the greatest integer not greater than, {all numbers}
Glencoe Algebra 2
12. has the same color eyes as, {all members of the Cleveland Symphony Orchestra} yes
11.
10. is the square of, {all numbers} no; reflexive, symmetric, transitive
9. is a multiple of, {all integers} no; symmetric
8. ⊥, {all lines in a plane} no; reflexive, transitive
7. is the spouse of, {all people in Roanoke, Virginia} no; reflexive, transitive
6. , {all polygons in a plane} yes
5. is a factor of, {all nonzero integers} no; symmetric
4. , {all numbers} no; symmetric
3. is the sister of, {all women in Tennessee} no; reflexive
2. , {all triangles in a plane} yes
1. , {all numbers} no; reflexive, symmetric
In each of the following, a relation and a set are given. Write yes if the
relation is an equivalence relation on the given set. If it is not, tell
which of the properties it fails to exhibit.
Answers
3. INCOME Manuel takes a job translating
English instruction manuals to Spanish.
He will receive $15 per page plus $100
per month. Manuel would like to work for
3 months during the summer and make
at least $1,500. Write and solve an
inequality to find the minimum number
of pages Manuel must translate in order
to reach his goal.
P D
7
2. PARTY FAVORS Janelle would like to
give a party bag to every person who is
coming to her party. The cost of the
party bag is $7 per person. Write an
inequality that describes the number
of people P that she can invite if
Janelle has D dollars to spend on
the party bags.
For all elements a and b of set A, if
a R b, then b R a.
Symmetric Property
Equality on the set of all real numbers is reflexive, symmetric, and transitive.
Therefore, it is an equivalence relation.
For any element a of set A, a R a.
Reflexive Property
A relation R on a set A is an equivalence relation if it has the following properties.
8:03 AM
5 < —2x — 7
12 < —2x
—6 < x
Enrichment
Equivalence Relations
1-5
____________ PERIOD _____
5/19/06
b 20
4. FINDING THE ERROR The sample
below shows how Brandon solved
5 2x 7. Study his solution and
determine if it is correct. Explain your
reasoning.
Word Problem Practice
NAME ______________________________________________ DATE
A2-01-873971
1. PANDAS An adult panda bear will eat
at least 20 pounds of bamboo every day.
Write an inequality that expresses this
situation.
1-5
NAME ______________________________________________ DATE______________ PERIOD _____
Lesson 1-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A19
(Lesson 1-5)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
Solving Compound and Absolute Value Inequalities
Lesson Reading Guide
11:30 P.M.
5:00 P.M.
Jason and Juanita
1
2
3
4
5
A20
2
3
4
5
{x x 1 or x 3}
Chapter 1
42
Glencoe Algebra 2
quantity is followed by a or symbol, the expression inside the
absolute value bars must be between two numbers, so this becomes an
and inequality. The number line graph will show a single interval
between two numbers. If the absolute value quantity is followed by a or symbol, it becomes an or inequality, and the graph will show two
disconnected intervals with arrows going in opposite directions.
5. Many students have trouble knowing whether an absolute value inequality should be
translated into an and or an or compound inequality. Describe a way to remember which
of these applies to an absolute value inequality. Also describe how to recognize the
difference from a number line graph. Sample answer: If the absolute value
Remember What You Learned
4. Write a statement equivalent to ⏐3x 7⏐ 8 that does not use the absolute value
symbol. 8 3x 7 8
3. Write a statement equivalent to ⏐4x 5⏐ 2 that does not use the absolute value
symbol. 4x 5 2 or 4x 5 2
1
5 4 3 2 1 0
1
2
3
3
4
4
5
5
2
4
and
6
8
2x 5 19
2x 14
x7
8 6 4 2 0
3y 2 7
3y 9
y3
2
or
or
or
4
2
4
5
7
8
9 11 13 15 17 19
6
Chapter 1
0
2
4
6
8 10 12 14 16
{w 4 w 14}
7. 22 6w 2 82
0 10 20 30 40 50 60 70 80
{y 29 y 54}
5. 100 5y 45 225
3
{x 7 x 15}
3. 18 4x 10 50
8 6 4 2 0
2
3
1
3
4
2
3
4
12
0
12
24
43
4 3 2 1 0
1
2
3
{all real numbers}
4
Glencoe Algebra 2
8. 4d 1 9 or 2d 5 11
24
{b b 12 or b 18}
6. b 2 10 or b 5 4
4 3 2 1 0
{k k 3 or k 2.5}
4. 5k 2 13 or 8k 1 19
10 0 10 20 30 40 50 60 70
{a a 5 or a 52}
{x 4 x 4}
1
4
2. 3a 8 23 or a 6 7
8
1. 10 3x 2 14
6
2y 1 9
2y 8
y 4
Example 2
Solve 3y 2 7 or
2y 1 9. Graph the solution set
on a number line.
The graph is the union of solution sets of
two inequalities.
Solve each inequality. Graph the solution set on a number line.
Exercises
8 6 4 2 0
4 x 7
3 2x 5
8 2x
4 x
Example 1
Solve 3 2x 5 19.
Graph the solution set on a number line.
Or
Compound
Inequalities
2
The graph is the intersection of solution sets of
two inequalities.
Answers
5 4 3 2 1 0
2. Use a compound inequality and set-builder notation to describe the following graph.
5 4 3 2 1 0
b. Graph the inequality that you wrote in part a on a number line.
1. a. Write a compound inequality that says, “x is greater than 3 and x is less than or
equal to 4.” 3 x 4
Juanita
Samir
1
Example: x 3 or x 1
5 4 3 2 1 0
Example: x 4 and x 3
8:03 AM
Read the Lesson
5:00 A.M.
1:30 A.M.
And
Compound
Inequalities
Compound Inequalities A compound inequality consists of two inequalities joined by
the word and or the word or. To solve a compound inequality, you must solve each part
separately.
Solving Compound and Absolute Value Inequalities
Study Guide and Intervention
5/19/06
Ora
Jason
• Five patients arrive at a medical laboratory at 11:30 A.M. for a glucose tolerance test.
Each of them is asked when they last had something to eat or drink. Some of the patients
are given the test and others are told that they must come back another day. Each of the
patients is listed below with the times when they started to fast. (The P.M. times refer to
the night before.) Which of the patients were accepted for the test?
Read the introduction to Lesson 1-6 in your textbook.
1-6
NAME ______________________________________________ DATE______________ PERIOD _____
A2-01-873971
Get Ready for the Lesson
1-6
NAME ______________________________________________ DATE______________ PERIOD _____
Lesson 1-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A20
(Lesson 1-6)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
(continued)
Solving Compound and Absolute Value Inequalities
Study Guide and Intervention
4
6
8
8 6 4 2 0
2
4
6
8
4
3
A21
⏐
1
2
3
4
8 12 16 20 24
2
4
6
8
40
20
0
20
40
4. ⏐a 9⏐ 30 {a a 39 or a 21}
8 6 4 2 0
2. ⏐4s⏐ 1 27 {s s 6.5 or s 6.5}
2
4
6
8 10 12
1
2
3
4
5
6
7
3
2
8
Chapter 1
9. ⏐4b 11⏐ 17 b b 7
0
7. ⏐10 2k⏐ 2 {k 4 k 6}
4 2 0
⏐
⏐
2
4
6
8
44
40
5 10 15 20 25 30
2
3
Glencoe Algebra 2
10. ⏐100 3m⏐ 20 m m 26 or m 10 5 0
8. 5 2 10 {x x 6 or x 26}
x
2
8 6 4 2 0
5. ⏐2f 11⏐ 9 {f f 1 or f 10} 6. ⏐5w 2⏐ 28 {w 6 w 5.2}
8 4 0
3. 3 5 {c 4 c 16}
c
⏐2
5 4 3 2 1 0
2
3
4
1
2
3
4
4 3 2 1 0
4 3 2 1 0
1
1
2
2
3
3
4
4
n 3
n 1
4
6
8 6 4 2 0
8.
6.
4 3 2 1 0
4 3 2 1 0
1
1
2
2
2
4
3
3
6
1
2
3
4
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
1
2
3
4
Chapter 1
4 2 0
2
4
6
8 10 12
19. ⏐n 5⏐ 7 {n 2 n 12}
4 3 2 1 0
17. ⏐7r⏐ 14 {r r 2 or r 2}
4 3 2 1 0
15. ⏐t⏐ 3 {t t 3 or t 3}
0
13. 8 3x 2 23 {x 2 x 7}
4 3 2 1 0
11. 10 5x 5 {x 2 x 1}
4 3 2 1 0
or c 0}
9. 2c 1 5 or c 0 {c c 2
8
4
4
8
n 2.5
n 4
1
2
3
4
1
2
3
4
or a 3}
1
2
3
4
numbers
1
1
2
2
3
3
4
4
45
2
4
6
8
Answers
8 6 4 2 0
Glencoe Algebra 2
20. ⏐h 1⏐ 5 {h h 6 or h 4}
4 3 2 1 0
18. ⏐p 2⏐ 2 4 3 2 1 0
16. ⏐6x⏐ 12 {x 2 x 2}
4 3 2 1 0
14. w 4 10 or 2w 6 all real
4 3 2 1 0
12. 4a 8 or a 3 {a a 2
4 3 2 1 0
10. 11 4y 3 1 {y 2 y 1}
Solve each inequality. Graph the solution set on a number line.
7.
5.
2
4. all numbers between 6 and 6 n 6
8 6 4 2 0
Write an absolute value inequality for each graph.
4 3 2 1 0
3. all numbers less than 1 or greater
than 1 n 1
1
2. all numbers less than 5 and greater
than 5 n 5
Answers
1. ⏐3x 4⏐ 8 x 4 x Solve each inequality. Graph the solution set on a number line.
Exercises
2
By statement 1 above, if ⏐2x 1⏐ 5, then
5 2x 1 5. Adding 1 to all three parts
of the inequality gives 4 2x 6.
Dividing by 2 gives 2 x 3.
By statement 2 above, if ⏐x 2⏐ 4, then
x 2 4 or x 2 4. Subtracting 2
from both sides of each inequality gives
x 2 or x 6.
8 6 4 2 0
Solve 2x 1 5.
Graph the solution set on a number line.
Example 2
Solve x 2 4. Graph
the solution set on a number line.
4 3 2 1 0
1. all numbers greater than or equal to 2
or less than or equal to 2 n 2
8:03 AM
Example 1
These statements are also true for and .
1. If ⏐a⏐ b, then b a b.
2. If ⏐a⏐ b, then a b or a b.
5/19/06
For all real numbers a and b, b 0, the following statements are true.
Use the definition of absolute value to rewrite an
absolute value inequality as a compound inequality.
Solving Compound and Absolute Value Inequalities
Skills Practice
Write an absolute value inequality for each of the following. Then graph the
solution set on a number line.
1-6
NAME ______________________________________________ DATE______________ PERIOD _____
A2-01-873971
Absolute Value Inequalities
1-6
NAME ______________________________________________ DATE______________ PERIOD _____
Lesson 1-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A21
(Lesson 1-6)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Chapter 1
20
10
0
10
20
n 10
4.
4 3 2 1 0
1
2
3
4
8 12 16 20 24 28 32
2
4
6
8 10 12 14
4
3
4
{x x 2
or x 8}
6. 3(5x 2) 24 or 6x 4 4 5x
2 0
3
8
n 2
4
2
3
4
5
5
A22
1
2
3
4
2
4
6
8 10 12 14 16
1
2
1
2
3
3
4
4
1
2
3
4
1
2
3
4
1
2
3
4
1
5
4 3 2 1 0
1
2
3
4
n
16. ⏐3n 2⏐ 2 1 n 3
3
4 3 2 1 0
14. ⏐x⏐ x 1 all real numbers
4 3 2 1 0
1
5
z
12. ⏐2z 2⏐ 3 z 2
2
8 7 6 5 4 3 2 1 0
10. ⏐y 5⏐ 2 {x 7 x 3}
4 3 2 1 0
Chapter 1
46
v 14.5 0.08; {v 14.42 v 14.58}
Glencoe Algebra 2
18. MANUFACTURING A company’s guidelines call for each can of soup produced not to vary
from its stated volume of 14.5 fluid ounces by more than 0.08 ounces. Write and solve an
absolute value inequality to describe acceptable can volumes.
r 24 6.5; {r r 17.5 or r 30.5}
17. RAINFALL In 90% of the last 30 years, the rainfall at Shell Beach has varied no more
than 6.5 inches from its mean value of 24 inches. Write and solve an absolute value
inequality to describe the rainfall in the other 10% of the last 30 years.
4 3 2 1 0
15. ⏐3b 5⏐ 2 4 3 2 1 0
13. ⏐2x 2⏐ 7 5 {x 2 x 0}
0
11. ⏐x 8⏐ 3 {x x 5 or x 11}
4 3 2 1 0
or w 9. ⏐2w⏐ 5 w w 2
2
1
Chapter 1
20N 50,000 100,000 and N 10,000; The attendance must be
between 7,500 and 10,000 people,
inclusive.
3. CONCERT Jacinta is organizing a large
fund-raiser concert in a space with a
maximum capacity of 10,000 people. Her
goal is to raise at least $100,000. Tickets
cost $20 per person. Jacinta spends
$50,000 to put the event together. Write
and solve a compound inequality that
describes N, the number of attendees
needed to achieve Jacinta’s goal.
n b 2n
2. HIKING For a hiking trip, everybody
must bring at least one backpack.
However, because of space limitations,
nobody is allowed to bring more than
two backpacks. Let n be the number of
people going on the hiking trip and b be
the number of backpacks allowed. Write
a compound inequality that describes
how b and n are related.
45 d < 55
47
Glencoe Algebra 2
s 12 and 3s 45; s is at least 12
and at most 15
7. Write a compound inequality for s using
parts A and B. Find the maximum and
minimum values for s.
s 12
6. A passenger needs to bring a soil sample
on the plane that is at least 1 cubic foot.
The passenger is bringing it in a suitcase
that is in the shape of a cube with side
length s inches. Write an inequality that
gives the minimum length for s.
h w 45
w
5. Write an inequality that describes the
airline’s carry-on size limitation.
l
h
An airline company has a size limitation for
carry-on luggage. The limitation states that
the sum of the length, width, and height of
the suitcase must not exceed 45 inches.
AIRLINE BAGGAGE For Exercises 5–7,
use the following information.
110 b 110
Answers
4 3 2 1 0
3}
7. 2x 3 15 or 3 7x 17 {x x 2} 8. 15 5x 0 and 5x 6 14 {x x 0
5. 8 3y 20 52 {y 4 y 24}
4
1
2
8:03 AM
Solve each inequality. Graph the solution set on a number line.
3.
4 3 2 1 0
8 6 4 2 0
4. NUMBERS Amy is thinking of two
numbers a and b. The sum of the two
numbers must be within 10 units of zero.
If a is between 100 and 100, write a
compound inequality that describes the
possible values of b.
Solving Compound and Absolute Value Inequalities
Word Problem Practice
1. AQUARIUM The depth d of an
aquarium tank for dolphins satisfies
|d 50| 5. Rewrite this as a
compound inequality that does not
involve the absolute value function.
1-6
____________ PERIOD _____
5/19/06
Write an absolute value inequality for each graph.
2. all numbers between 1.5 and 1.5, including
1.5 and 1.5 n 1.5
1. all numbers greater than 4 or less than 4 n 4
6
Solving Compound and Absolute Value Inequalities
Practice
NAME ______________________________________________ DATE
A2-01-873971
Write an absolute value inequality for each of the following. Then graph the
solution set on a number line.
1-6
NAME ______________________________________________ DATE______________ PERIOD _____
Lesson 1-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A1-A23
Page A22
(Lesson 1-6)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
Enrichment
Chapter 1
Solve 2x 10.
x 5 and x 5.
x 6 or x 4
3
3x 18 or 3x 4
A23
Chapter 1
x 6 or x 10; disjunction
9. 8 x 2
x 12 or x 16; disjunction
48
Glencoe Algebra 2
x 1 and x 4; conjunction
10. 5 2x 3
x 16 and x 8; conjunction
8. 4
x 3 4 x
7. 1 7
2
6. 7 2x 5
x 1 and x 1; conjunction
1
4
x 2 or x 0; disjunction
4. x 1 1
x 15 and x 1; conjunction
2. x 7 8
x and x ; conjunction
1
2
5. 3x 1 x
x 2 and x 3; conjunction
3. 2x 5 1
x 6 or x 6; disjunction
1. 4x 24
Solve each inequality. Then write whether the solution is a conjunction or
disjunction.
Exercises
A compound sentence that combines two statements by the word or is
a disjunction.
x 6 or x true.
Answers
Answers
4
3
Every solution for the inequality is a replacement for x that makes either
Solve each inequality.
Example 2
Solve 3x 7 11.
3x 7 11 means 3x 7 11 or 3x 7 11.
A compound sentence that combines two statements by the word and is
a conjunction.
Every solution for 2x 10 is a replacement for x that makes both x 5
and x 5 true.
8:03 AM
Solve each inequality.
2 x 10 means 2x 10 and 2x 10.
Example 1
5/19/06
An absolute value inequality may be solved as a compound sentence.
A2-01-873971
Conjunctions and Disjunctions
1-6
NAME ______________________________________________ DATE______________ PERIOD _____
A1-A23
Page A23
(Lesson 1-6)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2