Download Chapter 10: Algebra: More Equations and Inequalities

Algebraic Thinking:
Linear and Nonlinear Functions
Focal Point
Analyze and represent
proportional and
nonproportional linear
relationships.
CHAPTER 10
Algebra: More Equations and
Inequalities
Use graphs, tables, and
algebraic representations to make
predictions and solve problems.
Make connections
among various representations of
a numerical relationship.
CHAPTER 11
Algebra: Linear Functions
Identify proportional or
nonproportional linear relationships
in problem situations and solve
problems.
Make connections among
various representations of a numerical
relationship.
CHAPTER 12
Algebra: Nonlinear Functions
and Polynomials
Understand that a function
can be described in a variety of ways.
524
Michael Newman/PhotoEdit
Math and Economics
Getting Down to Business How would you like to run your own
business? On this adventure, you’ll be creating your own company.
Along the way, you’ll come up with a company name, slogan, and
product to sell to your peers at school. You’ll research the cost of
materials, create advertisements, and calculate potential profits.
You’ll also survey your peers to find out what they would be willing
to pay for your product, analyze the data, and adjust your projected
profit model. You’re going to need your algebra tool kit to make this
company work, so let’s get down to business!
Log on to tx.msmath3.com to begin.
Unit 5 Algebraic Thinking: Linear and Nonlinear Functions
Michael Newman/PhotoEdit
525
Algebra: More Equations
and Inequalities
10
Knowledge
and Skills
•
Use graphs, tables, and
algebraic representations to
make predictions and solve
problems. TEKS 8.5
•
Make connections among
various representations
of a numerical relationship.
TEKS 8.4
Key Vocabulary
arithmetic sequence (p. 544)
equivalent expressions (p. 528)
like terms (p. 529)
two-step equation (p. 534)
Real-World Link
Beaches The Texas shoreline has been decreasing at
an average rate of about 6 feet per year. You can write
an equation to describe the amount of shoreline for a
given number of years.
Algebra: More Equations and Inequalities Make this Foldable to help you organize your notes.
Begin with a plain sheet of 11” × 17” paper.
1 Fold in half lengthwise.
2 Fold again from top to
bottom.
3 Open and cut along the
second fold to make two
tabs.
4 Label each tab as shown.
%QUATIONS
)NEQUALIT
IES
526
Chapter 10 Algebra: More Equations and Inequalities
Terry Donnelly/Donnelly Austin Photography
GET READY for Chapter 10
Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2
Take the Online Readiness Quiz at tx.msmath3.com.
Option 1
Take the Quick Quiz below. Refer to the Quick Review for help.
Determine whether each statement
is true or false. (Used in Lesson 10-7)
Example 1
1. 10 > 4
2. 3 < -3
Determine whether the statement -2 > 1
is true or false.
3. -8 < -7
4. -1 > 0
Plot the points on a number line.
5. WEATHER The temperature in
Sioux City, Iowa, was -7°F while
the temperature in Des Moines,
Iowa, was -5°F. Which city was
warmer? Explain.
Write an algebraic equation for each
verbal sentence. (Used in Lessons 10-1 through
Since -2 is to the left of 1, -2 < 1. The
statement is false.
Example 2
6. Ten increased by a number is -8.
Write an algebraic equation for the
verbal sentence twice a number increased
by 3 is -5.
7. The difference of -5 and 3x is 32.
Let x represent the number.
10-3 and 10-5)
8. Twice a number decreased by
twice a number increased by 3 is -5
+3
2x
4 is 26.
9. The sum of 9 and a number is 14.
= -5
So, the equation is 2x + 3 = -5.
10. MONEY Bianca has $1 less than
twice as much as her brother. If
her brother had $15, how much
money did Bianca have?
Solve each equation. Check your
solution. (Used in Lessons 10-2, 10-3, 10-5,
and 10-7)
11. n + 8 = -9
12. 4 = m + 19
13. -4 + a = 15
14. z - 6 = -10
15. 3c = -18
16. -42 = -6b
17.
w = -8
_
4
Example 3
Solve 44 = k - 7.
44 = k - 7
+
7= +7
______
51 = k
Write the equation.
Add 7 to each side.
Simplify.
18. 12 = _
r
-7
Chapter 10 Get Ready for Chapter 10
527
10-1
Simplifying Algebraic
Expressions
Main IDEA
Use the Distributive
Property to simplify
algebraic expressions.
Targeted TEKS 8.16
The student uses
logical reasoning to
make conjectures
and verify conclusions.
(B) Validate his/her
conclusions using
mathematical properties
and relationships.
Also addresses TEKS 8.16(A).
You can use algebra tiles to rewrite the algebraic expression 2(x + 3).
Double this amount
of tiles to represent
2(x + 3).
Represent x + 3
using algebra tiles.
1
x
1
x
1
1
1
Rearrange the tiles by
grouping together the
ones with the same shape.
1
x
1
1
x
x
1
1
1
1
1
1
1
1. Choose two positive and one negative value for x. Then evaluate
2(x + 3) and 2x + 6 for each of these values. What do you notice?
NEW Vocabulary
equivalent expressions
term
coefficient
like terms
constant
simplest form
simplifying the expression
2. Use algebra tiles to rewrite the expression 3(x - 2). (Hint: Use one
green x-tile and 2 red –1-tiles to represent x - 2.)
In Chapter 1, you learned that expressions like 2(4 + 3) can be rewritten
using the Distributive Property and then simplified.
2(4 + 3) = 2(4) + 2(3)
Distributive Property
= 8 + 6 or 14
Multiply. Then add.
The Distributive Property can also be used to simplify an algebraic
expression like 2(x + 3).
2(x + 3) = 2(x) + 2(3)
Distributive Property
= 2x + 6
Multiply.
The expressions 2(x + 3) and 2x + 6 are equivalent expressions,
because no matter what x is, these expressions have the same value.
Write Expressions With Addition
Use the Distributive Property to rewrite each expression.
2 (y + 2)5
1 4(x + 7)
4(x + 7) = 4(x) + 4(7)
READING
in the Content Area
For strategies in reading
this lesson, visit
tx.msmath3.com.
528
= 4x + 28
(y + 2)5 = y · 5 + 2 · 5
Simplify.
a. 6(a + 4)
Chapter 10 Algebra: More Equations and Inequalities
b. (n + 3)8
= 5y + 10
c. -2(x + 1)
Simplify.
Write Expressions with Subtraction
Look Back You can
review multiplying
integers in
Lesson 1-6.
Use the Distributive Property to rewrite each expression.
3 6(p - 5)
Rewrite p - 5 as p + (-5).
6(p - 5) = 6[p + (-5)]
= 6(p) + 6(-5)
Distributive Property
= 6p + (-30)
Simplify.
= 6p - 30
Definition of subtraction
4 -2(x - 8)
-2(x - 8) = -2[x + (-8)]
Rewrite x - 8 as x + (-8).
= -2(x) + (-2)(-8)
Distributive Property
= -2x + 16
Simplify.
d. 3(y - 10)
e. -7(w - 4)
f. (n - 2)(-9)
When plus or minus signs separate an algebraic expression into parts,
each part is called a term. The numerical factor of a term that contains a
variable is called the coefficient of the variable.
This expression has three terms.
-2x + 16 + x
1 is the coefficient of x
- 2 is the coefficient of x
Like terms contain the same variables, such as 2x and x. A term without
a variable is called a constant. Constant terms are also like terms.
Vocabulary Link
Constant
Everyday Use unchanging
Rewriting a subtraction expression using addition will help you identify
the terms of an expression that contains subtraction.
Math Use a numeric term
without a variable
Identify Parts of an Expression
5 Identify the terms, like terms, coefficients, and constants in the
expression 6n - 7n - 4 + n.
6n - 7n - 4 + n = 6n + (-7n) + (-4) + n
Definition of subtraction
= 6n + (-7n) + (-4) + 1n Identity Property; n = 1n
• Terms: 6n, -7n, -4, n
• Like terms: 6n, -7n, n
• Coefficients: 6, -7, 1
• Constants: -4.
Identify the terms, like terms, coefficients,
and constants in each expression.
g. 9y - 4 - 11y + 7
Extra Examples at tx.msmath3.com
h. 3x + 2 - 10 - 3x
Lesson 10-1 Simplifying Algebraic Expressions
529
An algebraic expression is in simplest form if it has no like terms and
no parentheses. You can use the Distributive Property to combine like
terms. This is called simplifying the expression.
Simplify Algebraic Expressions
6 Simplify the expression 3y + y.
Equivalent
Expressions
To check whether
3y + y and 4y
are equivalent
expressions, substitute
any value for y and
see whether the
expressions have
the same value.
3y and y are like terms.
Identity Property; y = 1y
3y + y = 3y + 1y
= (3 + 1)y or 4y
Distributive Property; simplify.
7 Simplify the expression 7x - 2 - 7x + 6.
7x and -7x are like terms. -2 and 6 are also like terms.
7x - 2 - 7x + 6 = 7x + (-2) + (-7x) + 6
Definition of subtraction
= 7x + (-7x) + (-2) + 6
Commutative Property
= [7 + (-7)]x + (-2) + 6
Distributive Property
= -0x + 4
Simplify.
= 0 + 4 or 4
0x = 0 · x or 0
Simplify each expression.
i. 4z - z
j. 6 - 3n + 3n
k. 2g - 3 + 11 - 8g
8 FOOD At a baseball game, you buy some hot dogs that cost $3 each
and the same number of soft drinks for $2.50 each. Write an
expression in simplest form that represents the total amount spent.
Words
Real-World Link
In a recent year,
Americans were
expected to eat
26.3 million hot dogs
in major league
ballparks. This is
enough to stretch
from Dodger Stadium
in Los Angeles to the
Pirates’ PNC Stadium
in Pittsburgh.
Variable
$3 each some number
and
of hot dogs
for
$2.50 the same number
each for
of drinks
Let x represent the number of hot dogs or drinks.
3·x
Expression
+
2.50 · x
Simplify the expression.
3x + 2.50x = (3 + 2.50)x
= 5.50x
Distributive Property
Simplify.
The expression $5.50x represents the total amount spent.
Source: www.hot-dog.org
l. MONEY You have saved some money. Your friend has saved $50
less than you. Write an expression in simplest form that represents
the total amount of money you and your friend have saved.
Personal Tutor at tx.msmath3.com
530
Chapter 10 Algebra: More Equations and Inequalities
DiMaggio/Kalish/CORBIS
Examples 1–4
(pp. 528–529)
Example 5
(p. 529)
Examples 6, 7
(p. 530)
Example 8
(p. 530)
(/-%7/2+ (%,0
For
Exercises
16–27
28–33
34–39
40–43
See
Examples
1–4
5
6, 7
8
Use the Distributive Property to rewrite each expression.
1. 5(x + 4)
2. 2(n + 7)
3. (y + 6)3
4. (a + 9)4
5. 2(p - 3)
6. 6(4 - k)
7. -6(g - 2)
8. -3(a + 9)
Identify the terms, like terms, coefficients, and constants in each expression.
9. 5n - 2n - 3 + n
10. 8a + 4 - 6a - 5a
11. 7 - 3d - 8 + d
Simplify each expression.
12. 8n + n
13. 7n + 5 - 7n
14. 4p - 7 + 6p + 10
15. MOVIES You buy 2 drinks that each cost x dollars and a large bag of
popcorn for $3.50. Write an expression in simplest form that represents
the total amount of money you spent.
Use the Distributive Property to rewrite each expression.
16. 3(x + 8)
17. -8(a + 1)
18. (b + 8)5
19. (p + 7)(-2)
20. 4(x - 6)
21. 6(5 - q)
22. -8(c - 8)
23. -3(5 - b)
24. (d + 2)(-7)
25. -4(n - 3)
26. (10 - y)(-9)
27. (6 + z) 3
Identify the terms, like terms, coefficients, and constants in each expression.
28. 2 + 3a + 9a
29. 7 - 5x + 1
30. 4 + 5y - 6y + y
31. n + 4n - 7n - 1
32. -3d + 8 - d - 2
33. 9 - z + 3 - 2z
Simplify each expression.
34. n + 5n
35. 12c - c
36. 5x + 4 + 9x
37. 2 + 3d + d
38. -3r + 7 - 3r - 12
39. -4j - 1 - 4j + 6
Write an expression in simplest form that represents the total amount in
each situation.
40. SHOPPING You buy x shirts that each cost $15.99, the same number of
jeans for $34.99 each, and a pair of sneakers for $58.99.
41. PHYSICAL EDUCATION Each lap around the school track is a distance of
1
y yards. You ran 2 laps on Monday, 3_
laps on Wednesday, and 100 yards
2
on Friday.
42. FUND-RAISING You have sold t tickets for a school fund-raiser. Your friend
has sold 24 less than you.
43. BIRTHDAYS Today is your friend’s birthday. She is y years old. Her brother
is 5 years younger.
Lesson 10-1 Simplifying Algebraic Expressions
531
44. GOVERNMENT In 2005, in the Texas Legislature, there were 119 more
members in the House of Representatives than in the Senate. If there were
m members in the Senate, write an expression in simplest form to represent
the total number of members in the Texas Legislature.
45.
FIND THE DATA Refer to the Texas Data File on pages 16–19. Choose
some data and write a real-world problem in which you would write and
simplify an algebraic expression.
Use the Distributive Property to rewrite each expression.
46. 3(2y + 1)
47. -4(3x + 5)
48. -6(12 - 8n)
49. 4(x - y)
50. -2(3a - 2b)
51. (-2 - n)(-7)
52. 5x(y - z)
53. -6a(2b + 5c)
Simplify each expression.
54. -_ a - _ + _ a - _
2
5
1
4
7
10
1
5
55. 6p - 2r - 13p + r
56. -n + 8s - 15n - 22s
57. SCHOOL You are ordering T-shirts with your school’s mascot printed on
them. Each T-shirt costs $4.75. The printer charges a set-up fee of $30 and
$2.50 to print each shirt. Write two expressions that you could use to
represent the total cost of printing n T-shirts.
GEOMETRY Write two equivalent expressions for the area of each figure.
58.
12
59.
10
60.
x4
x7
x5
16
61. SCHOOL You spent m minutes studying on Monday. On Tuesday, you
studied 15 more minutes than you did on Monday. Wednesday, you
studied 30 minutes less than you did on Tuesday. You studied twice
as long on Thursday as you did on Monday. On Friday, you studied
20 minutes less than you did on Thursday. Write an expression in
simplest form to represent the number of minutes you studied for these
five days.
%842!02!#4)#%
See pages 720, 737.
Self-Check Quiz at
tx.msmath3.com
H.O.T. Problems
62. OPEN ENDED Write an expression that has four terms and simplifies
to 3n + 2. Identify the coefficient(s) and constant(s) in your expression.
63. Which One Doesn’t Belong? Identify the expression that is not equivalent to
the other three. Explain your reasoning.
x - 3 + 4x
5(x - 3)
6 + 5x - 9
5x - 3
64. CHALLENGE Simplify the expression 8x 2 - 2x + 12x - 3. Show that your
answer is true for x = 2.
65.
*/ -!4( Is 2(x - 1) + 3(x - 1) = 5(x - 1) a true statement?
(*/
83 *5*/(
If so, justify your answer using mathematical properties. If not, give
a counterexample.
532
Chapter 10 Algebra: More Equations and Inequalities
66. Olinda purchased 2 shirts for x dollars
67. Which expression can be used to
each, a pair of shoes for $59.99, and
lunch for $3.50. What is an expression
in simplest form that represents
the total amount in dollars that
Olinda spent?
represent the perimeter of the figure?
4x
x3
x3
6x
A 63.49 - 2x
B 2x + 63.49
C 2x + 59.99 - 3.50
F 10x + 9
H 16x + 6
G 12x + 6
J
24x + 9
D 2x - 63.49
TECHNOLOGY For Exercises 68 and 69, refer to the graphs.
Graph A
Graph B
$6$0LAYER3ALES
$6$0LAYER3ALES
Choose an appropriate type of display for each
situation. (Lesson 9-7)
70. the amount of each flavor of ice cream sold
relative to the total sales
players sold in 2003 was the number of DVD
players sold in 2002?
3ALESMILLIONS
69. About what percent of the number of DVD
3ALESMILLIONS
68. Which graph gives the impression that the
number of DVD players sold in 2002 was
about 20% the amount sold in 2003?
(Lesson 9-8)
9EAR
9EAR
71. the number of people attending a fair in
specific age ranges
Find each probability.
(Lesson 8-4)
72. tossing a coin 3 times and getting heads each time
73. rolling a number cube 3 times and getting a number greater than
4 each time
74. PARKS A circular fountain in a park has a diameter of 8 feet. The park
director wants to build a fountain whose area is four times that of the
current fountain. What will be the diameter of the new fountain?
(Lesson 7-1)
PREREQUISITE SKILL Solve each equation. Check your solution.
(Lessons 1-9 and 1-10)
75. x + 8 = 2
76. y - 5 = -9
77. 32 = -4n
78.
_a = -15
3
Lesson 10-1 Simplifying Algebraic Expressions
533
10-2
Solving Two-Step Equations
Main IDEA
Solve two-step equations.
Targeted TEKS 8.5
The student uses
graphs, tables,
and algebraic
representations to make
predictions and solve
problems. (A) Predict, find,
and justify solutions to
application problems using
appropriate tables, graphs,
and algebraic equations.
NEW Vocabulary
two-step equation
BOOK SALE Linda bought four books
at a book sale benefiting a local charity.
The handwritten receipt she received
was missing the cost for the hardback
books she purchased.
I]Vc`Ndj[dg
NdjgHjeedgi
(]VgYWVX`h &eVeZgWVX`h &
1. Explain how you could use the work
IdiVaeV^Y ,
backward strategy to find the cost of
each hardback book. Then find the cost.
The solution to this problem can also
be found by solving the equation
3x + 1 = 7, where x is the cost per
hardback book. This equation can be
modeled using algebra tiles.
1
x
x
x
3x 1
1
1
1
1
1
1
1
7
A two-step equation contains two operations. In the equation
3x + 1 = 7, x is multiplied by 3 and then 1 is added. To solve two-step
equations, undo each operation in reverse order.
Solve Two-Step Equations
1 Solve 3x + 1 = 7.
METHOD 1
Use a model.
Remove one 1-tile from the mat.
1
x
x
x
3x 1 1
1
1
1
1
1
1
1
71
Separate the remaining tiles
into 3 equal groups.
Use the Subtraction Property
of Equality.
3x + 1 = 7
- 1 =-1
____________
3x
= 6
x
x
3x
1
1
1
1
1
1
3x = 6
3
534
Aaron Haupt
Chapter 10 Algebra: More Equations and Inequalities
3
x=2
6
There are 2 1-tiles in each
group, so the solution is 2.
Write the equation.
Subtract 1
from each side.
Use the Division Property
of Equality.
3x
6
_
=_
x
Use symbols.
METHOD 2
Divide each side by 3.
Simplify.
BrainPOP® tx.msmath3.com
_
2 Solve 25 = 1 n - 3.
4
METHOD 1
METHOD 2
Vertical method
1
25 = _
n-3
4
_
25 = 1 n - 3
4
+3=
+3
____________
1
28 = _
n
4
_1 n - 3 + 3 = 25 + 3
4
_1 n = 28
1
4 · 28 = 4 · _
n
4
4
1
4·_
n = 4 · 28
Simplify.
4
112 = n
_1 n - 3 = 25
Write the equation.
Add 3 to each side.
Horizontal method
4
n = 112
Multiply each side
by 4.
The solution is 112.
Solve each equation. Check your solution.
a. 3x + 2 = 20
b. 5 + 2n = -1
c. -1 = _ a + 9
1
2
Personal Tutor at tx.msmath3.com
Some two-step equations have a term with a negative coefficient.
Equations with Negative Coefficients
3 Solve 6 - 3x = 21.
6 - 3x = 21
Common Error
A common mistake
when solving the
equation in Example
3 is to divide each
side by 3 instead of
-3. Remember that
you are dividing by
the coefficient of the
variable, which in this
instance is a negative
number.
Write the equation.
6 + (-3x) = 21
Rewrite the left side as addition.
6 - 6 + (-3x) = 21 - 6
Subtract 6 from each side.
-3x = 15
Simplify.
-3x
15
_
=_
Divide each side by -3.
-3
-3
x = -5
Simplify.
The solution is -5.
Check
6 - 3x = 21
Write the equation.
6 - 3(- 5) 21
Replace x with -5.
6 - (-15) 21
Multiply.
6 + 15 21
21 = 21
To subtract a negative number, add its opposite.
✓
The sentence is true.
Solve each equation. Check your solution.
d. 10 - 7p = 52
Extra Examples at tx.msmath3.com
e. -19 = -3x + 2
f.
n
_
- 2 = -18
-3
Lesson 10-2 Solving Two-Step Equations
535
Sometimes it is necessary to combine like terms before solving
an equation.
Combine Like Terms First
4 Solve -2y + y - 5 = 11. Check your solution.
-2y + y - 5 = 11
Write the equation.
Identity Property; y = 1y
-2y + 1y - 5 = 11
-y - 5 = 11
Combine like terms; -2y + 1y = (-2 + 1)y or -y.
-y - 5 + 5 = 11 + 5
Add 5 to each side.
-y = 16
Simplify.
-1y
16
_
=_
-1
-y = -1y; divide each side by -1.
-1
y = -16
Simplify.
The solution is –16.
Check
-2y + y - 5 = 11
Write the equation.
Replace y with -16.
-2(-16) + (-16) - 5 11
32 + (-16) - 5 11
11 = 11
Multiply.
✓
The statement is true.
Solve each equation. Check your solution.
g. x + 4x = 45
Examples 1–3
(pp. 534–535)
h. 10 = 2a + 13 - a
i. -3 = 6 - 5w + 2w
Solve each equation. Check your solution.
1. 6x + 5 = 29
4.
_2 x - 5 = 7
3
2. -2 = 9m - 11
3. 10 = _ + 3
5. 3 - 5y = -37
6.
a
4
c
_
-4=3
-2
Example 3
7. ELECTRONICS Mr. Sampson bought a home theater system. The total cost of
(p. 535)
the system was $816, and he pays $34 a month on the balance. The current
balance owed is $272. Solve the equation 272 = 816 - 34m to determine the
number of monthly payments Mr. Sampson has made.
Example 4
(p. 536)
Solve each equation. Check your solution.
8. 6k - 10k = 16
9. 5d + 4 - 6d = 11
10. 1 = 13 - 2p + 4p
11. MOVIES Cassidy went to the movies with some of her friends. The tickets
cost $6.50 apiece, and each person received a $1.75 student discount.
The total amount paid for all the tickets was $33.25. Solve the equation
33.25 = 6.50p - 1.75p to determine the number of people who went to
the movies.
536
Chapter 10 Algebra: More Equations and Inequalities
(/-%7/2+ (%,0
For
Exercises
12–19,
24, 25
20–23
26–33
See
Examples
1, 2
3
4
Solve each equation. Check your solution.
12. 2h + 9 = 21
13. 11 = 2b + 17
14. 5 = 4a - 7
15. -17 = 6p - 5
16. 2g - 3 = -19
17. 16 = 5x - 9
g
18. 13 = _ + 4
3
y
19. 5 + _ = -3
8
1
22. -_x - 7 = -11
2
21. 13 - 3d = -8
20. 3 - 8c = 35
23. 15 - _ = 28
w
4
24. SCHOOL TRIP At an amusement park, each student is given $19 for food.
This covers the cost of 2 meals at x dollars each plus $7 worth of snacks.
Solve 2x + 7 = 19 to find how much money the school expects each student
will spend per meal.
25. SHOPPING Suppose you receive a $75 online gift to your favorite music site.
You want to purchase some CDs that cost $14 each. There will be a $5
shipping and handling fee. Solve 14n + 5 = 75 to find the number of CDs
you can purchase.
Solve each equation. Check your solution.
26. 28 = 3m - 7m
27. y + 5y = 24
28. 3 - 6x + 8x = 9
29. -21 = 9a - 15 - 3a
30. 26 = g + 10 - 3g
31. 8x + 5 - x = -2
32. GAMES Brent had $26 when he went to the fair. After playing 5 games and
then 2 more, he had $15.50 left. Solve 15.50 = 26 - 5p - 2p to find the price
for each game.
33. SPORTS LaTasha paid $75 to join a summer golf program. The course
where she plays charges $30 per round, but since she is a student, she
receives a $10 discount per round. If LaTasha spent $375, use the equation
375 = 30g - 10g + 75 to find out how many rounds of golf LaTasha played.
Solve each equation. Check your solution.
34. 4(x + 2) = 20
37.
a-4
_
= 12
5
35. 6(w - 2) = 54
38.
36. -27 = 3(t + 1)
n+3
_
= -4
39.
8
6+z
_
= -2
10
14 ft
40. HOME IMPROVEMENT If Mr. Arenth wants to
put new carpeting in the room shown, how
many square feet should he order?
41. TEMPERATURE The formula F = 1.8C + 32
%842!02!#4)#%
See pages 720, 737.
6c 8 ft
can be used to convert between degrees
Fahrenheit (°F) and degrees Celsius (°C).
What is the temperature in degrees Celsius
if the current temperature is 77°F?
5 3c ft
25
42. GEOMETRY Write an equation to
Self-Check Quiz at
tx.msmath3.com
−−
represent the length of AB. Then find
the value of x.
13
x
2x
A
Lesson 10-2 Solving Two-Step Equations
B
537
H.O.T. Problems
43. FIND THE ERROR Alexis and Tomás are solving the equation
2x + 7 = 16. Who is correct? Explain.
2x + 7 = 16
2x + 7 - 7 = 16 - 7
2x = 9
9
2x
_
=_
2
2
2x + 7 = 16
16
2x
_
+7=_
2
2
x+7=8
x+7-7=8-7
x=1
x = 4.5
Alexis
Tomás
44. CHALLENGE Solve (x + 5) 2 = 49. (Hint: There are two solutions.)
45.
*/ -!4( Explain how you can use the work backward problem(*/
83 *5*/(
solving strategy to solve a two-step equation.
46. Ms. Peak’s total monthly charge for
47. A whirlpool holds 1,600 gallons of
local and long-distance telephone
service c can be found using the
equation c = 25 + 0.04m, where m
represents the number of minutes of
long-distance calls she made. Find the
number of minutes of long-distance
calls made for a month in which the
total charge was $29.40.
water. The water has been draining
from the pool at a rate of 80 gallons
per hour. Currently, there are 640
gallons of water left in the pool. Use
the equation 640 = 1,600 - 80t to
determine how long the pool has been
draining.
F 8 hours
H 16 hours
A 110 minutes
C 735 minutes
G 12 hours
J
B 176 minutes
D 1,360 minutes
Use the Distributive Property to rewrite each expression.
48. 6(a + 6)
49. -3(x + 5)
January through July. Which graph
might be misleading? Explain. (Lesson 9-8)
(Lesson 10-1)
50. (y - 8)4
52. SALES The graphs show Jin’s sales for
28 hours
51. -8(p - 7)
Sales
$900
y
Sales
$600
$600
$500
$300
$400
0
J F M A M J J x
Month
0
y
J F M A M J J x
Month
PREREQUISITE SKILL Write an algebraic equation for each verbal sentence.
(Lesson 1-7)
53. A number increased by 5 is 17.
538
Chapter 10 Algebra: More Equations and Inequalities
(l)RubberBall/Alamy Images, (r)CORBIS
54. The quotient of a number and 2 is -2.
10-3
Writing Two-Step Equations
Main IDEA
Write two-step equations
that represent real-life
situations.
Targeted TEKS 8.4
The student
makes connections
among various
representations of a
numerical relationship. The
student is expected to
generate a different
representation of data given
another representation of
data (such as a table, graph,
equation, or verbal
description). Also addresses
TEKS 8.14(D).
HOME ENTERTAINMENT Your parents
offer to loan you the money to buy
a $600 sound system. You give them
$125 as a down payment and agree
to make monthly payments of
$25 until you have repaid the loan.
Payments
Amount Paid
0
125 + 25(0) = $125
1
125 + 25(1) = $150
2
125 + 25(2) = $175
3
125 + 25(3) = $200
1. Let n represent the number of
payments. Write an expression
that represents the amount of the loan paid after n payments.
2. Write and solve an equation to find the number of payments
you will have to make in order to pay off your loan.
3. What type of equation did you write for Exercise 2? Explain
your reasoning.
In Chapter 1, you learned how to write verbal sentences as one-step
equations. Some verbal sentences translate to two-step equations.
Words
The sum of 125 and 25 times a number is 600.
Variable
Let n represent the number.
Equation
125 + 25n = 600
Translate Sentences into Equations
Translate each sentence into an equation.
Sentence
Equation
1 Eight less than three times a number is -23.
3n - 8 = -23
2 Thirteen is 7 more than twice a number.
13 = 2n + 7
3 The quotient of a number and 4, decreased by 1,
n
_
-1=5
is equal to 5.
4
Translate each sentence into an equation.
a. Fifteen equals three more than six times a number.
b. If 10 is increased by the quotient of a number and 6, the result is 5.
c. The difference between 12 and twice a number is 18.
Extra Examples at tx.msmath3.com
Lesson 10-3 Writing Two-Step Equations
539
4 FUND-RAISING Your Class Council needs $600. With only $210 in the
treasury, they decide to raise the rest by selling donuts for a profit
of $1.50 per dozen. How many dozen will they need to sell?
Treasury
amount
Words
plus
1.50 per dozen sold
$600.
Let d represent the number of dozens.
Variable
210
Equation
+
d
1.50 ·
210 + 1.50d = 600
Real-World Career
How Does a FundRaising Professional
Use Math?
Fund-raising
professionals use
equations to help set
and meet fund-raising
goals.
equals
210 - 210 + 1.50d = 600 - 210
=
Write the equation.
Subtract 210 from each side.
1.50d = 390
Simplify.
1.50d
390
_
=_
Divide each side by 1.50.
1.50
1.50
600
d = 260
They need to sell 260 dozen.
5 DINING You and your friend’s lunch totaled $19. Your lunch cost $3
For more information,
go to tx.msmath3.com.
more than your friend’s. How much was your friend’s lunch?
Your friend’s lunch
Words
your lunch equals
$19.
Let f represent the cost of your friend’s lunch.
Variable
Look Back
You can review
writing equations
in Lesson 1–7.
plus
Equation
f
f + f + 3 = 19
Write the equation.
2f + 3 = 19
2f + 3 - 3 = 19 - 3
+
19
Subtract 3 from each side.
Simplify.
2f
16
_
=_
Divide each side by 2.
2
=
Combine like terms.
2f = 16
2
f+3
f=8
Your friend spent $8.
d. METEOROLOGY Suppose the current temperature is 54°F. It is
expected to rise 2°F each hour for the next several hours. In how
many hours will the temperature be 78°F?
e. GEOMETRY The perimeter of a rectangle is 40 inches. The width is
8 inches shorter than the length. Write and solve an equation to
find the dimensions of the rectangle.
Personal Tutor at tx.msmath3.com
540
Chapter 10 Algebra: More Equations and Inequalities
Jon Feingersch/CORBIS
Examples 1–3
(p. 539)
Translate each sentence into an equation.
1. One more than three times a number is 7.
2. Seven less than twice a number is -1.
3. The quotient of a number and 5, less 10, is 3.
For Exercises 4 and 5, write and solve an equation to solve each problem.
Example 4
(p. 540)
Example 5
(p. 540)
(/-%7/2+ (%,0
For
Exercises
6–9
10–13
14, 15
See
Examples
1–3
4
5
4. BOOK FINES You return a book that is 5 days overdue. Including a previous
unpaid overdue balance of $1.30, your new balance is $2.05. How much is
the daily fine for an overdue book?
5. SHOPPING Marty paid $121 for shoes and clothes. He paid $45 more for
clothes than he did for shoes. How much did Marty pay for the shoes?
Translate each sentence into an equation.
6. Four less than five times a number is equal to 11.
7. Fifteen more than twice a number is 9.
8. Eight more than four times a number is -12.
9. Six less than seven times a number is equal to -20.
For Exercises 10–15, write and solve an equation to solve each problem.
10. PERSONAL FITNESS Angelica joins a local
gym called Fitness Solutions. If she sets aside
$1,000 in her annual budget for gym costs, use
the ad at the right to determine how many
hours she can spend with a personal trainer.
11. VACATION While on vacation, you purchase
4 identical T-shirts for some friends and a watch
for yourself, all for $75. You know that the watch
cost $25. How much did each T-shirt cost?
Annual Membership: $720
Personal Trainers Available
($35/h)
12. PHONE SERVICE A telephone company advertises long distance service for
7¢ per minute plus a monthly fee of $3.95. If your bill for one month was
$12.63, find the number of minutes you used making long distance calls.
13. VIDEO GAMES You and two of your friends share the cost of renting a
video game system for 5 nights. Each person also rents one video game
for $6.33. If each person pays $11.33, what was the cost of renting the
video game system?
14. MONUMENTS From ground level to the tip of the torch, the Statue of Liberty
and its pedestal are 92.99 meters high. The pedestal is 0.89 meter higher
than the statue. How high is the Statue of Liberty?
Lesson 10-3 Writing Two-Step Equations
541
15. GEOMETRY Find the value of x in the
x˚
parallelogram at the right.
134˚
134˚
x˚
ANIMALS For Exercises 16 –18, use the information at the left.
16. The top speed of a peregrine falcon is 20 miles per hour less than three
times the top speed of a cheetah. What is the cheetah’s top speed?
17. A sailfish can swim up to 1 mile per hour less than one fifth the top speed
of a peregrine falcon. Find the top speed that a sailfish can swim.
18. The peregrine falcon can reach speeds about 14 miles per hour more than
Real-World Link
When diving, the
peregrine falcon can
reach speeds of up to
175 miles per hour.
Source: Time for
Kids Almanac
7 times the speed of the fastest human. What is the approximate top speed
of the fastest human?
19. BASKETBALL In a basketball game, 2 points are awarded for making a
regular basket, and 1 point is awarded for making a foul shot. Emeril
scored 21 points during one game. Three of those points were for foul
shots. The rest were for regular goals. Find the number of regular baskets
that Emeril made during the game.
20. SKIING In aerial skiing competitions,
the total judges’ score is multiplied by the
jump’s degree of difficulty and then added
to the skier’s current score to obtain their
final score. After her second jump,
Martin’s final score is 216.59. The
degree of difficulty for Toshiro’s second
jump is 4.45. What must the judges’ score
for Toshiro’s jump be in order for her to tie
Martin for first place?
Skier
Score
Martin, S.
100.23
Toshiro, M.
105.34
Moseley, K.
93.99
Long, A.
87.50
Cruz, P.
80.63
Thompson, L.
75.23
21. ALGEBRA Three consecutive even
integers can be represented by n, n + 2,
and n + 4. If the sum of three consecutive
even integers is 36, what are the integers?
%842!02!#4)#%
See pages 720, 737.
JOBS For Exercises 22 and 23, use the following information.
Hunter and Amado are each trying to save $600 for a summer trip. Hunter
started with $150 and earns $7.50 per hour working at a grocery store. Amado
has nothing saved, but he earns $12 per hour painting houses.
22. Make a conjecture about who will take longer to save enough money for
Self-Check Quiz at
tx.msmath3.com
H.O.T. Problems
the trip. Justify your reasoning.
23. Write and solve two equations to check your conjecture.
24. OPEN ENDED Write two different statements that translate into the same
two-step equation.
25. CHALLENGE Student Council has $200 to divide among the top class
finishers in a used toy drive. Second place will receive twice as much as
third place. First place will receive $15 more than second place. Write and
solve an equation to find how much each winning class will receive.
542
Chapter 10 Algebra: More Equations and Inequalities
(l)Tim Fitzharris/Masterfile, (r)Cris Cole/Allsport/Getty Images
26. SELECT A TECHNIQUE Sherrie bought 3 bottles of sports drink for $6.42. If the
sales tax was $0.42, which technique would you use to determine the cost
of each bottle of sports drink? Justify your selection. Then find the cost of
each bottle of sports drink.
mental math
estimation
paper/pencil
*/ -!4( Write about a real-world situation that can be solved
(*/
83 *5*/(
27.
using a two-step equation. Then write the equation and solve the problem.
28. A company employs 72 workers.
29. As a salesperson, Ms. Anderson
It plans to increase the number of
employees by 6 per month until it has
twice its current workforce. Which
equation can be used to determine m,
the number of months it will take for
the number of employees to double?
receives a weekly base salary of $325
plus 7% of her weekly sales. At the
end of one week, she earned $500.
Which equation can be used to find
her sales s for that week?
A 6m + 72m = 144
G 325 + 7s = 500
B 2m + 72 = 144
H 325 + 0.7s = 500
C 2(6m + 72) = 144
J
F 325s + 0.07 = 500
325 + 0.07s = 500
D 6m + 72 = 144
Solve each equation. Check your solution.
30. 5x + 2 = 17
(Lesson 10-2)
31. -7b + 13 = 27
Simplify each expression.
34. 5x + 6 - x
32. -6 = _ + 1
33. -15 = -4p + 9
36. 7a - 7a - 9
37. 3 - 4y + 9y
n
8
(Lesson 10-1)
35. 8 - 3n + 3n
38. UTILITIES An airport has changed the carrels used for
public telephones. The old carrels consisted of four
sides of a rectangular prism. The new carrels are half
of a cylinder with an open top. How much less
material is needed to construct a new carrel than
an old carrel? (Lesson 7-7)
PREREQUISITE SKILL Find each rate of change.
39.
CDS
Cost ($)
3
5
8
37.50
62.50
100.00
Old Design
New Design
45 in.
45 in.
13 in.
26 in.
26 in.
(Lesson 4-9)
40.
DayS
10
15
20
Height (cm)
8
11
14
Lesson 10-3 Writing Two-Step Equations
543
10-4
Sequences
Main IDEA
Write algebraic
expressions to determine
any term in an arithmetic
sequence.
Targeted TEKS 8.5
The student uses
graphs, tables, and
algebraic
representations to make
predictions and solve
problems. (A) Predict, find,
and justify solutions to
application problems using
appropriate tables, graphs,
and algebraic equations.
(B) Find and evaluate an
algebraic expression to
determine any term in an
arithmetic sequence (with a
constant rate of change).
Consider the following pattern.
Number Of Toothpicks
2 triangles
3 triangles
3 toothpicks
5 toothpicks
7 toothpicks
1. Continue the pattern for 4, 5, and 6 triangles. How many
toothpicks are needed for each case?
2. How many additional toothpicks are needed for 4 triangles? How
many total toothpicks will you need for 7 triangles?
The number of toothpicks needed for each pattern form a sequence. A
sequence is an ordered list of numbers. Each number in the list is called
a term. An arithmetic sequence is a sequence in which the difference
between any two consecutive terms is the same.
3,
5,
7,
9, 11, …
+2 +2 +2 +2
NEW Vocabulary
sequence
term
arithmethic sequence
common difference
1 triangle
Number of Triangles
The difference is called the
common difference.
To find the next number in an arithmetic sequence, add the common
difference to the last term.
Identify Arithmetic Sequences
READING Math
And So On Three dots
following a list of numbers
are read as and so on.
1 State whether the sequence 17, 12, 7, 2, -3, ... is arithmetic. If it is,
state the common difference and write the next three terms.
17, 12, 7,
-5 -5 -5
2, -3
Notice that 12 - 17 = -5, 7 - 12 = -5, and so on.
-5
The terms have a common difference of -5, so the sequence is
arithmetic. Continue the pattern to find the next three terms.
-3, -8, -13, -18
-5
-5
The next three terms are -8, -13, and -18.
-5
State whether each sequence is arithmetic. Write yes or no. If it is,
state the common difference and write the next three terms.
a. 2, 6, 10, 14, 18, …
544
Chapter 10 Algebra: More Equations and Inequalities
b. -4, -8, -16, -32, …
Arithmetic sequences can be described by writing an algebraic
expression.
Describe an Arithmetic Sequence
2 Write an expression that can be used to find the nth term of the
sequence 4, 8, 12, 16, ... . Then write the next three terms.
Use a table to examine
the sequence.
+1 +1 +1
Term Number (n)
The terms have a common
Term
difference of 4. Also, each
term is 4 times its term
number. An expression that
can be used to find the nth term is 4n.
1
2
3
4
4
8
12
16
+4 +4 +4
The next three terms are 4(5) or 20, 4(6) or 24, and 4(7) or 28.
Write an expression that can be used to
find the nth term of each sequence. Then find the next three terms.
c. -2, -4, -6, -8, …
d.
_1 , _1 , _1 , _2 , …
e. 0.5, 1, 1.5, 2, …
6 3 2 3
3 TEXT MESSAGING The table shows the monthly cost of sending text
messages. How much would it cost to send 25 text messages?
The common difference between
the costs is 0.25. This implies that
the expression for the nth message
sent is 0.25n. Compare each cost to
the value of 0.25n for each number
of messages.
Real-World Link
Approximately 45% of
teens use a cell phone
and 33% use text
messaging.
Source: The Pew Internet
& American Life Project
Messages
Cost ($)
1
5.25
2
5.50
3
5.75
4
6.00
+1
+1
+1
Each cost is 5 more than 0.25n. So,
the expression 0.25n + 5 is the cost
of n messages. To find the cost of
sending 25 messages, let c represent
the cost. Then write and solve an
equation for n = 25.
c = 0.25n + 5
Write the equation.
c = 0.25(25) + 5
Replace n with 25.
c = 6.25 + 5 or 11.25
Simplify.
+0.25
+0.25
+0.25
Messages
Cost ($)
0.25n
1
5.25
0.25
2
5.50
0.50
3
5.75
0.75
4
6.00
1.00
It would cost $11.25 to send 25 text messages.
Write and solve an expression to find the
nth term of each arithmetic sequence.
f. 4, 9, 14, 19, … ; n = 12
Extra Examples at tx.msmath3.com
Bob Daemmrich/The Image Works
g. -20, -16, -12, -8, … ; n = 20
Lesson 10-4 Sequences
545
4 Which expression can be used to find the nth term in the following
sequence, where n represents a number’s position in the sequence?
Position in Sequence
1
2
3
4
n
Term
3
5
7
9
?
A n+2
C 2n + 1
B 2n
D 3n
Read the Test Item You need to find an expression to describe
any term.
Solve the Test Item The terms have a common difference of 2 for
every 1 increase in positive number. So the expression contains 2n.
Eliminate
Possibilities
Test n = 1 in each
expression. Since
2(1) ≠ 3, choice B is
eliminated. Next test
n = 2. Since 2 + 2 ≠ 5
and 3(2) ≠ 5, choices
A and D are eliminated.
So the answer is
choice C.
• Eliminate choices A and D because they do not contain 2n.
• Eliminate choice B because 2(1) ≠ 3.
• The expression in choice C is correct for all the listed terms. So the
correct answer is C.
h. Let n represent the position of a number in the sequence _ , _ , _ , 1, ….
1 1 3
4 2 4
Which expression can be used to find any term in the sequence?
1
F n+_
1
H _
n
G 2n
4
J 4n
4
Personal Tutor at tx.msmath3.com
Example 1
(p. 544)
State whether each sequence is arithmetic. Write yes or no. If it is, state the
common difference. Write the next three terms of the sequence.
2. 11, 4, -2, -7, -11, ...
1. 2, 4, 6, 8, 10, ...
Example 2
(p. 545)
Write an expression that can be used to find the nth term of each sequence.
Then write the next three terms of the sequence.
5. -5, -10, -15, -20, … 6.
4. 3, 6, 9, 12, …
Example 3
(p. 545)
Example 4
(p. 546)
3 _
1 _
_
, 1, _
, 2, …
10 5 10 5
Write and solve an equation to find the nth term of each arithmetic sequence.
7. 25, 23, 21, 19, ... ; n = 8
9.
8. 3, 10, 17, 24, ... ; n = 25
TEST PRACTICE In the sequence below, which expression can be used to
find the value of the term in the nth position?
Position
1
2
3
4
5
n
Value of Term
6
7
8
9
10
?
A n+1
546
3. 8, 2, -4, -10,-16, ...
B n+5
Chapter 10 Algebra: More Equations and Inequalities
C 2n
D 6n
(/-%7/2+ (%,0
For
Exercises
10–15
16–19
22–29
20, 21,
39, 40
See
Examples
1
2
3
4
State whether each sequence is arithmetic. Write yes or no. If it is, state the
common difference. Write the next three terms of the sequence.
10. 20, 24, 28, 32, 36, …
12.
11. 1, 10, 100, 1,000, 10,000, …
1
189, 63, 21, 7, 2_
,…
13. -6, -4, -2, 0, 2, …
3
15. 4, 6_, 9, 11_, 14, …
1
2
14. 1, 2, 5, 10, 17, …
1
2
Write an expression that can be used to find the nth term in each sequence.
Then write the next three terms of the sequence.
16. 2, 4, 6, 8, …
19.
_2 , _4 , 1_1 , 1_3 , …
5 5
5
5
_1 , _2 , 1, 1_1 , …
17. 12, 24, 36, 48, …
18.
20. 5, 9, 13, 17, …
21. 1, 4, 7, 10, …
3 3
3
Write and solve an equation to find the nth term of each arithmetic sequence.
22. 3, 7, 11, 15, … ; n = 8
23. 23, 25, 27, 29, … ; n = 12
24. 10, 5, 0, -5, … ; n = 21
25. 27, 19, 11, 3, … ; n = 17
26. 34, 49, 64, 79, … ; n = 200
27. 52, 64, 76, 88, … ; n = 102
FITNESS For Exercises 28 and 29, use the table.
Week
28. If Luther continues the pattern shown in the
Time Jogging
(min)
1
8
2
16
3
24
4
32
5
?
ˆ}ÕÀiÊÓ
ˆ}ÕÀiÊÎ
table, how many minutes will he spend jogging
each day during his fifth week of jogging?
29. Is the amount of time Luther jogs proportional
to the number of weeks he has jogged? Explain.
GEOMETRY For Exercises 30 and 31, use the
figure at the right.
30. How many squares will be in Figure 18?
31. Is the number of squares in each figure
proportional to the number of the figure?
Explain.
ˆ}ÕÀiÊ£
SKIING For Exercises 32–34, use the following information.
A ski resort offers a one-day lift pass for $40 and a yearly lift pass for $400.
Number of Visits
1
2
Total Cost with One Day Passes ($)
40
80
400
400
Total Cost with Yearly Pass ($)
%842!02!#4)#%
See pages 721, 737.
3
4
5
32. Is the sequence formed by the total cost with one day passes arithmetic?
Explain.
33. Is the sequence formed by the total cost with a yearly pass arithmetic?
Explain.
Self-Check Quiz at
tx.msmath3.com
34. How many times would a person have to go skiing to make the yearly pass
a better buy?
Lesson 10-4 Sequences
547
H.O.T. Problems
35. OPEN ENDED Write an arithmetic sequence in which the common difference
1
is -_
.
3
36. REASONING Determine whether the following statement is always, sometimes
or never true. Explain.
A sequence in which a number is added to a term in order to find the next
term is an arithmetic sequence.
37. CHALLENGE Write an expression that can be
used to find the nth term of sequence.
38.
Position
1
3
5
7
Term
8
14
20
26
*/ -!4( Write a real-world problem which uses an arithmetic
(*/
83 *5*/(
sequence. Then solve your problem.
39. In the sequence below, which
40. The expression below describes a
expression can be used to find the
value of the term in the nth position?
Position
1
2
3
4
5
n
-12 - 4(n - 1)
If n represents a number’s position
in the sequence, which pattern of
numbers does the expression describe?
Value of Term
0.6
1.2
1.8
2.4
3.0
?
A n - 0.4
C
_3 n
n
B _
D
n + 0.6
5
pattern of numbers.
F -12, -16, -20, -24, -28, …
G -12, -8, -4, 0, 4, …
H 12, 8, 4, 0, -4, …
J
12, 16, 20, 24, 28, …
5
41. SHOPPING Marisa bought 4 paperback books, all at the same price. The
tax on her purchase was $2.35, and the total was $34.15. What was the
price of each book? (Lesson 10-3)
Solve each equation.
(Lesson 10-2)
42. 9 + 5y = 19
43. -6 = 4 + 2x
44. 8 - k = 17
45. 2 = 18 - 4d
46. READING Rosa checked out three books from the library. While she was at
the library, she visited the fiction, nonfiction, and biography sections.
What are the possible combinations of book types she could have
checked out? (Lesson 8-2)
PREREQUISITE SKILL Simplify each expression.
47. 2x - 8 + 2x
548
48. -5n + 7 + 5n
Chapter 10 Algebra: More Equations and Inequalities
(Lesson 10-1)
49. 8p - 3 + 3
50. -6 - 15a + 6
Extend
10-4
Main IDEA
Determine numbers that
make up the Fibonacci
Sequence.
Math Lab
The Fibonacci Sequence
Leonardo of Pisa, also known as Fibonacci, created story problems
based on a series of numbers that became known as the Fibonacci
sequence. In this lab, you will investigate this sequence.
Targeted TEKS 8.16
The student uses
logical reasoning to
make conjectures
and verify conclusions. (A)
Make conjectures from
patterns or sets of examples
and nonexamples.
Using grid paper or dot paper, draw
a “brick” that is 2 units long and 1
unit wide. If you build a “walkway”
of grid paper bricks, there is only
one way to build a walkway that is
1 unit wide.
1 unit
Using two bricks, draw all of
the different walkways that are
2 units wide. There are two ways
to build the walkway.
2 units
Using three bricks,
draw all of the different
walkways that are
3 units wide. There
are three ways to build
the walkway.
3 units
Draw all of the different walkways that are 4 units, 5 units,
and 6 units long using the brick.
Copy and complete the table.
Length of Walkway
0
1
2
3
Number of Ways to
Build the Walkway
1
1
2
3
4
5
6
ANALYZE THE RESULTS
1. Explain how each number is related to the previous numbers in the
pattern.
2. Tell the number of ways there are to build a walkway that is 8 units
long. Do not draw a model.
3. MAKE A CONJECTURE Write a rule describing how you generate
numbers in the Fibonacci sequence.
4. RESEARCH Use the Internet or other resource to find how the
Fibonacci sequence is related to nature, music, or art.
Extend 10-4 Math Lab: The Fibonacci Sequence
549
CH
APTER
10
Mid-Chapter Quiz
Lessons 10-1 through 10-4
Use the Distributive Property to rewrite each
expression. (Lesson 10-1)
1. 3(x + 2)
2. -2(a - 3)
3. 5(3c - 7)
4. -4(2n + 3)
Translate each sentence into an equation. Then
find each number. (Lesson 10-3)
16. Nine more than the quotient of a number
and 3 is 14.
17. The quotient of a number and -7, less 4,
Simplify each expression.
is -11.
(Lesson 10-1)
5. 2a - 13a
6. 6b + 5 - 6b
7. 2m + 5 - 8m
8. 7x + 2 - 8x + 5
18. The difference between three times a
number and 10 is 17.
19. The difference between twice a number
and 13 is -21.
9. Identify the terms, like terms, coefficients,
and constants in the expression 5 - 4x +
x - 3. (Lesson 10-1)
20. MOVING A rental company charges $52
per day and $0.32 per mile to rent a moving
van. Ms. Misel was charged $202.40 for a
3-day rental. How many miles did she
drive? (Lesson 10-3)
Solve each equation. Check your solution.
(Lesson 10-2)
10. 3m + 5 = 14
11. -2k + 7 = -3
12. 11 = _a + 2
13. -15 = -7 - p
1
3
14.
TEST PRACTICE A diagram of a room is
shown below.
State whether each sequence is arithmetic.
Write yes or no. If it is, state the common
difference. Write the next three terms of the
sequence. (Lesson 10-4)
21. 13, 17, 21, 25, 29, …
w
22. 64, -32, 16, -8, 4, …
23. -5, -16, -25, -34, -43, ...
2w 3
If the perimeter of the room is 78 feet, find
its width. (Lesson 10-2)
24.
A 12 ft
B 15 ft
(Lesson 10-4)
C 25 ft
D 27 ft
15. EXERCISE Brandi rode her bike the same
distance on Tuesday and Thursday, and
20 miles on Saturday for a total of 50
miles for the week. Solve the equation
2m + 20 = 50 to find the distance Brandi
rode on Tuesday and Thursday. (Lesson 10-2)
550
TEST PRACTICE Which expression can
be used to find the nth term in the following
arithmetic sequence, where n represents a
number’s position in the sequence?
Position
1
2
3
4
n
Term
2
5
8
11
?
F 2n
G n+3
H 3n - 1
J
4n - 2
25. Find the 12th term of the arithmetic
Chapter 10 Algebra: More Equations and Inequalities
sequence 3, 7, 11, 15, … .
(Lesson 10-4)
Explore
10-5
Main IDEA
Algebra Lab
Equations with Variables
on Each Side
You can also use algebra tiles to solve equations that have variables on
each side of the equation.
Solve equations with
variables on each side
using algebra tiles.
Targeted TEKS 8.15
The student
communicates
about Grade 8
mathematics through
informal and mathematical
language, representations,
and models. (A)
Communicate mathematical
ideas using language,
efficient tools, appropriate
units, and graphical,
numerical, physical, or
algebraic mathematical
models.
Interactive Lab tx.msmath3.com
1 Use algebra tiles to solve 3x + 1 = x + 5.
1
x
x
x
1
3x 1
x
x
x
x
x
2x 1 1
x
1
1
1
1
Remove the same number of x-tiles
from each side of the mat until there
are x-tiles on the only one side.
Remove the same number of
1-tiles from each side of the mat
until the x-tiles are by themselves
on one side.
1
x
1
1
1
1
51
1
1
1
1
x
2x
1
xx5
1
x
1
Model the equation.
1
3x x 1
1
x5
1
x
1
Separate the tiles into two equal
groups.
4
Therefore, x = 2. Since 3(2) + 1 = 2 + 5, the solution is correct.
Use algebra tiles to solve each equation.
a. x + 2 = 2x + 1
b. 2x + 7 = 3x + 4
c. 2x - 5 = x - 7
d. 8 + x = 3x
e. 4x = x - 6
f. 2x - 8 = 4x - 2
ANALYZE THE RESULTS
1. Identify the property of equality that allows you to remove a 1-tile
or -1-tile from each side of an equation mat.
2. Explain why you can remove an x-tile from each side of the mat.
Explore 10-5 Algebra Lab: Equations with Variables on Each Side
551
2 Use algebra tiles to solve x - 4 = 2x + 2.
1 1
x
1 1
x4
1
x
x
1 1
xx4
x
x
1
Remove the same number of x-tiles
from each side of the mat until there
is an x-tile by itself on one side.
1
2x x 2
1 1 1
1 1 1
Model the equation.
2x 2
1 1
x
1
x
1
1
1
1
It is not possible to remove the same
number of 1-tiles from each side of
the mat. Add two -1-tiles to each side
of the mat.
4 (2) x 2 (2)
1 1 1
1 1 1
6
x
1
1
1
1
Remove the zero pairs from the right
side. There are six -1-tiles on the left
side of the mat.
x
Therefore, x = -6. Since -6 - 4 = 2(-6) + 2, the solution is correct.
Use algebra tiles to solve each equation.
g. x + 6 = 3x - 2
h. 3x + 3 = x - 5
i. 2x + 1 = x - 7
j. x - 4 = 2x + 5
k. 3x - 2 = 2x + 3
l. 2x + 5 = 4x - 1
ANALYZE THE RESULTS
3. Solve x + 4 = 3x - 4 by removing 1-tiles first. Then solve the
equation by removing x-tiles first. Does it matter whether you
remove x-tiles or 1-tiles first? Is one way more convenient? Explain.
4. MAKE A CONJECTURE In the set of algebra tiles, -x is represented by x .
Explain how you could use -x-tiles and other algebra tiles to
solve -3x + 4 = -2x - 1.
552
Chapter 10 Algebra: More Equations and Inequalities
10-5
Solving Equations with
Variables on Each Side
Main IDEA
Solve equations with
variables on each side.
Targeted TEKS 8.5
The student uses
graphs, tables, and
algebraic
representations to make
predictions and solve
problems. (A) Predict, find,
and justify solutions to
application problems using
appropriate tables, graphs,
and algebraic equations.
SPORTS You and your friend
are having a race. You give
your friend a 15-meter head
start. During the race, you
average 6 meters per second
and your friend averages
5 meters per second.
Time
(s)
Friend’s
Distance (m)
Your
Distance (m)
0
15 + 5(0) = 15
6(0) = 0
1
15 + 5(1) = 20
6(1) = 6
2
15 + 5(2) = 25
6(2) = 12
3
15 + 5(3) = 30
6(3) = 18
1. Copy the table. Continue
filling in rows to find how
long it will take you to catch up to your friend.
2. Write an expression for your distance after x seconds.
3. Write an expression for your friend’s distance after x seconds.
4. What is true about the distances you and your friend have gone
when you catch up to your friend?
5. Write an equation that could be used to find how long it will take
for you to catch up to your friend.
Some equations, like 15 + 5x = 6x, have variables on each side of the
equals sign. To solve these equations, use the Addition or Subtraction
Property of Equality to write an equivalent equation with the variables
on one side of the equals sign. Then solve the equation.
Equations with Variables on Each Side
1 Solve 15 + 5x = 6x. Check your solution.
15 + 5x = 6x
Write the equation.
15 + 5x - 5x = 6x - 5x
Subtract 5x from each side.
15 = x
Simplify by combining like terms.
Subtract 5x from the left side of the
equation to isolate the variable.
Subtract 5x from the right side of
the equation to keep it balanced.
To check your solution, replace x with 15 in the original equation.
Check
5x + 15 = 6x
Write the original equation.
5(15) + 15 6(15)
90 = 90
Replace x with 15.
✓ The sentence is true.
The solution is 15.
Extra Examples at tx.msmath3.com
Westlight Stock/OZ Production/CORBIS
Lesson 10-5 Solving Equations with Variables on Each Side
553
2 Solve 6n - 1 = 4n - 5.
6n - 1 = 4n - 5
Write the equation.
6n - 4n - 1 = 4n - 4n – 5 Subtract 4n from each side.
2n - 1 = -5
Simplify.
2n - 1 + 1 = -5 + 1
Add 1 to each side.
2n = -4
Simplify.
n = -2
Mentally divide each side by 2.
The solution is -2. Check this solution.
Solve each equation. Check your solution.
a. 8a = 5a + 21
b. 3x - 7 = 8x + 23
c. 7g - 12 = 3 + 2g
3 CELL PHONES A cellular phone provider charges $24.95 per month
plus $0.10 per minute for calls. Another cellular provider charges
$19.95 per month plus $0.20 per minute for calls. For how many
minutes of calls is the monthly cost of both providers the same?
Words
$24.95 per month plus
$0.10 per minute
equals
$19.95 per month plus
$0.20 per minute
Variable
Let m represent the minutes.
Equation
24.95 + 0.10m = 19.95 + 0.20m
24.95 + 0.10m = 19.95 + 0.20m
24.95 + 0.10m - 0.10m = 19.95 + 0.20m - 0.10m
24.95 = 19.95 + 0.10m
24.95 - 19.95 = 19.95 - 19.95 + 0.10m
5 = 0.10m
5
0.10m
_
=_
0.10
0.10
50 = m
Write the equation.
Subtract 0.10m
from each side.
Subtract 19.95 from
each side.
Divide each side
by 0.10.
The monthly cost is the same for 50 minutes of calls.
Real-World Link
Congress established
the first official
United States flag
on June 14, 1777.
Source: firstgov.gov
d. FLAGS The length of a flag is 0.3 foot less than twice its width.
If 17.4 feet of gold fringe is used along the perimeter of the
flag, find the dimensions of the flag.
Personal Tutor at tx.msmath3.com
554
Chapter 10 Algebra: More Equations and Inequalities
MPI/Getty Images
Examples 1, 2
(pp. 553–554)
Example 3
Solve each equation. Check your solution.
1. 5n + 9 = 2n
2. 3k + 14 = k
3. 10x = 3x - 28
4. 7y - 8 = 6y + 1
5. 2a + 21 = 8a - 9
6. -4p - 3 = 2 + p
7. CAR RENTAL EZ Car Rental charges $40 a day plus $0.25 per mile. Ace
(p. 554)
(/-%7/2+ (%,0
For
Exercises
8–11
12–19
20–23
See
Examples
1
2
3
Rent-A-Car charges $25 a day plus $0.45 per mile. What number of
miles results in the same cost for one day?
Solve each equation. Check your solution.
8. 7a + 10 = 2a
9. 11x = 24 + 8x
10. 9g - 14 = 2g
11. m - 18 = 3m
12. 5p + 2 = 4p - 1
13. 8y - 3 = 6y + 17
14. 15 - 3n = n - 1
15. 3 - 10b = 2b - 9
16. -6f + 13 = 2f - 11
17. 2z - 31 = -9z + 24
18. 2.5h - 15 = 4h
19. 21.6 - d = 5d
Define a variable, write an equation, and solve to find each number.
20. Eighteen less than three times a number is twice the number.
21. Eleven more than four times a number equals the number less 7.
For Exercises 22 and 23, write and solve an equation to solve each problem.
22. MOVIES For an annual membership fee of $30, you can join a movie club
that will allow you to purchase tickets for $5.50 each at your local theater.
If the theater in your area charges $8 for movie tickets, determine how
many movie tickets you will have to buy through the movie club for the
cost to equal that of buying tickets at the regular price.
23. FOOD DRIVES The seventh graders at your school have collected 345 cans
for the canned food drive and are averaging 115 cans per day. The eighth
graders have collected 255 cans, but vow to win the contest by collecting
an average of 130 cans per day. If both grades continue collecting at these
rates, after how many days will the number of cans they have collected
be equal?
Write an equation to find the value of x so that each pair of polygons has the
same perimeter. Then solve.
24.
x4
x1
x2
x3
%842!02!#4)#%
12x
25.
12x
12x
x5
12x
6x 9
x7
12x
x 10
See pages 721, 737.
26. GEOMETRY Write and solve an equation to
Self-Check Quiz at
tx.msmath3.com
find the perimeter and area of the square
at the right.
2x 8
4x 2
Lesson 10-5 Solving Equations with Variables on Each Side
555
27. CRAFT FAIRS The Art Club is selling handcrafted mugs at a local craft fair.
Vendors at the fair must pay $5 for a booth plus 10% of their sales. It costs
$8 in materials to make each mug. The club sells each mug for $10. Write
and solve an equation to find how many mugs they must sell to break
even.
H.O.T. Problems
28. OPEN ENDED Write an equation that has variables on each side with a
solution of 5.
3x 3
29. CHALLENGE Find the area of the
parallelogram at the right.
x3
*/ -!4( Explain how to solve
(*/
83 *5*/(
30.
5x 1
the equation 1 - 3x = 5x - 7.
31. Carpet cleaner A charges a fee of
32. Find the value of x so that the
$28.25 plus $18 a room. Carpet cleaner
B charges a fee of $19.85 plus $32 a
room. Which equation can be used to
find the number of rooms for which
the total cost of both carpet cleaners is
the same?
polygons have the same perimeter.
A 28.25x + 18 = 19.85x + 32
F 4
B 28.25 + 32x = 19.85 + 18x
G 3
C 28.25 + 18x = 19.85 + 32x
H 2
D (28.25 + 18)x = (19.85 + 32)x
J
2x
x4
2x
2x
2x
2x
x4
2x
x1
1
33. SAVINGS Nate has $10 in the bank. Each week he adds $2. How much
will he have after 20 weeks?
(Lesson 10-4)
34. SHOPPING At the market, Zacarias buys a bunch of grapes for $1.29 per
pound and a watermelon for $3.98. The total for his purchase was $7.85.
How many pounds of grapes did Zacarias buy? (Lesson 10-3)
A box contains 12 red tiles, 15 blue tiles, 10 yellow tiles, and 11 green tiles.
One tile is randomly chosen and not replaced. Then another tile is selected.
Find each probability. (Lesson 8-4)
35. P(red, yellow)
36. P(blue, green)
37. P(green, green)
38. P(yellow, blue)
39. PREREQUISITE SKILL Enrique has $37.50 to spend at the cinema. A drink
costs $1.75, popcorn costs $2.25, and tickets cost $8.50. Use the work
backward strategy to determine how many friends he can invite to go with
him if he pays for himself and for his friends. (Lesson 1-8)
556
Chapter 10 Algebra: More Equations and Inequalities
10-6 Problem-Solving Investigation
MAIN IDEA: Guess and check to solve problems.
Targeted TEKS 8.14 The student applies Grade 8 mathematics to solve problems connected to everyday experiences,
investigations in other disciplines, and activities in and outside of school. (C) Select or develop an appropriate problemsolving strategy from a variety of different types, including...systematic guessing and checking...to solve a problem.
e-Mail:
GUESS AND CHECK
YOUR MISSION: Solve the problem by guessing and
checking the solution
THE PROBLEM: Find the number of tickets collected
at the Balloon Pop and the Bean-Bag Toss.
Missy: We collected 150 tickets during
the Fall Carnival. It took 3 tickets to
play the Bean-Bag Toss and 2 tickets to
play the Balloon Pop. Ten more games were
played at the Bean-Bag Toss booth than at
the Balloon Pop.
EXPLORE
PLAN
SOLVE
CHECK
The Bean-Bag Toss was 3 tickets, and the Balloon Pop was 2 tickets. The number
of games played at the Bean-Bag Toss were 10 more than at the Balloon Pop.
Make a systematic guess and check to see if it is correct.
Find the combination that gives 150 total tickets. In the list, p is the number of
Balloon Pop games and t is the number of Bean-Bag Toss games.
p
t
2p + 3t
Check
12
22
2(12) + 3(22) = 90
too low
30
40
2(30) + 3(40) = 180
too high
27
37
2(27) + 3(37) = 165
still too high
24
34
2(24) + 3(34) = 150
correct
So 2(24) or 48 tickets were from the Balloon Pop and 3(34) or 102 tickets were
from the Bean-Bag Toss.
Thirty-four Balloon Pop games is 10 more than 24 Bean-Bag Toss games. Since
48 tickets plus 102 tickets is 150 tickets, the guess is correct.
1. Explain why it is important to make a systematic, organized list of your
guesses and their results when using the guess and check strategy.
2.
*/ -!4( Write a problem that could be solved by guessing
(*/
83 *5*/(
and checking. Then write the steps you would take to find the solution.
Lesson 10-6 Problem-Solving Investigation: Guess and Check
John Evans
557
9. RECREATION During a routine, ballet dancers
For Exercises 3–5, solve using the guess and
check strategy.
are evenly spaced in a circle. If the sixth
person is directly opposite the sixteenth
person, how many people are in the circle?
3. NUMBER THEORY A number is squared, and
the result is 576. Find the number.
ANALYZE TABLES For Exercises 10 and 11, use
the following information.
4. MONEY MATTERS Dominic has exactly $2
in quarters, dimes, and nickels. If he has
13 coins, how many of each coin does he
have?
5. GIFTS At a park souvenir shop, a mug costs
The school cafeteria surveyed 34 students
about their dessert preference. The results
are listed below.
$3, and a pin costs $2. Chase bought either a
mug or a pin for each of his 11 friends. If he
spent $30 on these gifts and bought at least
one of each type of souvenir, how many of
each did he buy?
Number of
Students
Preference of
Students
25
apples
20
oranges
15
bananas
2
all three
1
no fruit
15
apples or oranges
8
bananas or apples
3
oranges only
Use any strategy to solve Exercises 6–9. Some
strategies are shown below.
G STRATEGIES
PROBLEM-SOLVIN
tep plan.
• Use the four-s
m.
• Draw a diagra
10. How many students prefer only bananas?
• Make a table.
• Guess
11. How many do not prefer apples?
and check.
6. GEOMETRY The length of the rectangle
below is longer than its width w. List the
possible whole number dimensions for the
rectangle, and identify the dimensions that
give the smallest perimeter.
For Exercises 12–14, select the appropriate
operation(s) to solve the problem. Justify your
selection(s) and solve the problem.
12. TECHNOLOGY The average Internet user
2
A ⫽ 84 in
1
hours online each week. What
spends 6_
w
2
percent of the week does the average user
spend online?
ᐉ
7. DINING The cost of your meal is $8.25. If you
want to leave a 15% tip, would it be more
reasonable to expect the tip to be about $1.25
or about $1.50?
13. READING Terrence is reading a 255-page
book for his book report. He needs to read
twice as many pages as he has already read
to finish the book. How many pages has he
read so far?
8. DESIGN Edu-Toys is designing
a new package to hold a
set of 30 alphabet blocks
like the one shown. Give
two possible dimensions
for the box.
558
14. NUMBER SENSE Find the product of
2 in.
2 in.
2 in.
Chapter 10 Algebra: More Equations and Inequalities
1
1
1
1
1
1
,1-_
,1-_
,1-_
, ..., 1 - _
,1-_
,
1-_
2
2
1
and 1 - _
.
50
3
4
48
49
10-7
Inequalities
Main IDEA
Write and graph
inequalities.
Preparation for
TEKS A.1
The student
understands that
a function represents a
dependence of one quantity
on another and can be
described in a variety of
ways. (C) Describe functional
relationships for given
problem situations and write
equations or inequalities to
answer questions arising
from the situations.
SIGNS The top sign indicates that trucks more
than 10 feet 6 inches tall cannot pass. The other
sign indicates that a speed of 45 miles per hour
or less is legal.
1. Name three truck heights that can safely pass
on a road where the first sign is posted. Can a
truck that is 10 feet 6 inches tall pass? Explain.
2. Name three speeds that are legal according to
the second sign. Is a car traveling at 45 miles
per hour driving at a legal speed? Explain.
In Chapter 1, you learned that a mathematical sentence that contains
> or < is called an inequality. When used to compare a variable and
a number, inequalities can describe a range of values.
Write Inequalities with < or >
Write an inequality for each sentence.
1 SAFETY A package must
weigh less than 80 pounds.
Let w = package’s weight.
w < 80
2 AGE You must be over
55 years old to join.
Let a = person’s age.
a > 55
a. ROLLER COASTERS Riders must be taller than 48 inches.
b. SPORTS Members of a swim team must be under 15 years of age.
READING Math
Inequality Symbols
≤ less than or equal to
≥ greater than or equal to
The symbols ≤ or ≥ combine < or > with part of the equals sign.
Write Inequalities with ≤ or ≥
Write an inequality for each sentence.
3 VOTING You must be 18 years
of age or older to vote.
Let a = person’s age.
a ≥ 18
4 DRIVING Your speed must be
65 miles per hour or less.
Let s = car’s speed.
s ≤ 65
c. CARS A toddler must weigh at least 40 pounds to use a booster seat.
d. TRAVEL A fuel tank holds at most 16 gallons of gasoline.
Extra Examples at tx.msmath3.com
Doug Martin
Lesson 10-7 Inequalities
559
Inequalities
• is less than
• is fewer than
Words
• is greater than
• is more than
• exceeds
<
Symbols
• is less than or
equal to
• is no more than
• is at most
>
• is greater than or
equal to
• is no less than
• is at least
≤
≥
Inequalities with variables are open sentences. When the variable is
replaced with a number, the inequality may be true or false.
Determine the Truth of an Inequality
For the given value, state whether each inequality is true or false.
6 10 ≤ 7 - x, x = -3
5 a + 2 > 8, a = 5
Symbols Read 7 ≯ 8
as 7 is not greater
than 8.
a + 2 > 8 Write the inequality.
10 ≤ 7 - x
Write the inequality.
5 + 2 8 Replace a with 5.
10 7 - (-3)
Replace x with -3.
10 ≤ 10
Simplify.
7 ≯ 8 Simplify.
Since 7 is not greater than 8,
7 > 8 is false.
While 10 < 10 is false, 10 = 10
is true, so 10 ≤ 10 is true.
For the given value, state whether each inequality is true or false.
e. n - 6 < 15, n = 18
f. -3p ≥ 24, p = 8
g. -2 > 5y - 7, y = 1
Inequalities can be graphed on a number line. Since it is impossible to
show all the values that make an inequality true, an open or closed circle
is used to indicate where these values begin, and an arrow to the left or
to the right is used to show that they continue in the indicated direction.
Graph an Inequality
Graph each inequality on a number line.
8 n≥3
7 n<3
Place an open circle at 3.
Then draw a line and an
arrow to the left.
1
2
3
4
5
The open circle means
the number 3 is not
included in the graph.
Place a closed circle at 3.
Then draw a line and an
arrow to the right.
1
2
3
4
5
The closed circle means
the number 3 is
included in the graph.
Graph each inequality on a number line.
h. x > 2
i. x < 1
Personal Tutor at tx.msmath3.com
560
Chapter 10 Algebra: More Equations and Inequalities
j. x ≤ 5
k. x ≥ -4
Examples 1–4
(p. 559)
Write an inequality for each sentence.
1. RESTAURANTS Children under the age of 6 eat free.
2. TESTING A maximum of 45 minutes is given to complete section A.
Examples 5, 6
(p. 560)
Examples 7, 8
(p. 560)
(/-%7/2+ (%,0
For
Exercises
10–15
16–21
22–29
See
Examples
1–4
5, 6
7, 8
For the given value, state whether each inequality is true or false.
3. x - 11 < 9, x = 20
4. 42 ≥ 6a, a = 8
5.
n
_
+ 1 ≤ 6; n = 15
3
Graph each inequality on a number line.
6. n > 4
7. p ≤ 2
8. x ≥ 0
9. a < 7
Write an inequality for each sentence.
10. MOVIES Children under 13 are not permitted without an adult.
11. SHOPPING You must spend more than $100 to receive a discount.
12. ELEVATORS An elevator’s maximum load is 3,400 pounds.
13. FITNESS You must run at least 4 laps around the track.
14. GRADES A grade of no less than 70 is considered passing.
15. MONEY The cost can be no more than $25.
For the given value, state whether each inequality is true or false.
16. 12 + a < 20, a = 9
17. 15 - k > 6, k = 8
18. -3y < 21; y = 8
19. 32 ≤ 2x, x = 16
n
20. _ ≥ 5, n = 12
4
21.
-18
_
> 9, x = -2
x
Graph each inequality on a number line.
22. x > 6
23. a > 0
24. y < 8
25. h < 2
26. w ≤ 3
27. p ≥ 7
28. 1 ≤ n
29. 4 ≥ d
Jgfikj@eali`\j
SPORTS For Exercises 30 –33, use
the graph that shows the number
of children ages 5 –14 treated
recently in U.S. emergency rooms.
"ICYCLING
"ASKETBALL
30. In which sport(s) were more
31. In which sport(s) were at least
75,000 children injured?
32. Of the sports listed, which have
fewer than 100,000 injuries?
%842!02!#4)#% 33. Write an inequality comparing
See pages 722, 737.
the number treated for soccerrelated injuries with those
Self-Check Quiz at
treated for football-related
tx.msmath3.com
injuries.
3PORT
than 150,000 children injured?
&OOTBALL
"ASEBALL
3OFTBALL
3OCCER
3KATEBOARDING
.UMBEROF4REATED)NJURIESTHOUSANDS
Source: Children’s Hospital of Pittsburgh
Lesson 10-7 Inequalities
561
H.O.T. Problems
34. OPEN ENDED Write an inequality using ≤ or ≥. Then give a situation that
can be represented by the inequality.
35. FIND THE ERROR Valerie and Diego are writing an inequality for the
expression at least 2 hours of homework. Who is correct? Explain.
h≤2
h≥2
Valerie
Diego
36. CHALLENGE Determine whether the following statement is always,
sometimes, or never true. Explain your reasoning.
If x is a real number, then x ≥ x.
*/ -!4( If a < b and b < c, what is true about the relationship
(*/
83 *5*/(
37.
between a and c? Explain your reasoning and give examples using both
positive and negative values for a, b, and c.
38. Conner can spend no more than
4 hours at the swimming pool today.
Which graph represents the time that
Conner can spend at the pool?
A
39. What inequality is graphed below?
F x > -3
G x ≥ -3
H x < -3
B
J
x ≤ -3
C
D
40. SOUVENIRS The Alamo gift shop sells regular postcards in packages of 5
and large postcards in packages of 3. If Román bought 16 postcards, how
many packages of each did he buy? (Lesson 10-6)
41. CAR RENTAL Suppose you can rent a car for either $35 a day plus $0.40 a
mile or for $20 a day plus $0.55 per mile. Write and solve an equation to
find the number of miles that result in the same cost for one day. (Lesson 10-5)
562
Chapter 10 Algebra: More Equations and Inequalities
(l)Robin Lynne Gibson/Getty Images, (r)Richard Hutchings/Photo Researchers
CH
APTER
10
Study Guide
and Review
Download Vocabulary
Review from tx.msmath3.com
Key Vocabulary
arithmetic sequence
Be sure the following
Key Concepts are noted
in your Foldable.
%QUATIONS
)NEQUALIT
IES
Key Concepts
Algebraic Expressions
(Lesson 10-1)
• Like terms contain the same variables.
(p. 544)
sequence (p. 544)
simplest form (p. 530)
coefficient (p. 529)
common difference
(p. 544)
simplifying the
expression (p. 530)
term of expression (p. 529)
constant (p. 529)
term of sequence (p. 544)
equivalent expressions
two-step equation (p. 534)
(p. 528)
like terms (p. 529)
• A constant is a term without a variable.
• An algebraic expression is in simplest form if it
has no like terms and no parentheses.
Equations
(Lessons 10-1, 10-2, and 10-5)
• To solve a two-step equation, undo each
operation in reverse order.
• To solve equations with variables on each side of
the equals sign, use the Addition or Subtraction
Property of Equality to write an equivalent
equation with the variables on one side of the
equals sign. Then solve the equation.
Vocabulary Check
State whether each sentence is true or false.
If false, replace the underlined word or
number to make a true sentence.
1. Like terms are terms that contain different
variables.
2. A two-step equation is an equation that
contains two operations.
3. A coefficient is a term without an
Sequences
(Lesson 10-4)
• A sequence is an ordered list of numbers.
• Each number in the list is called a term.
• An arithmetic sequence is a sequence in which
the difference between any two consecutive terms
is the same.
Inequalities
(Lesson 10-7)
• When used to compare a variable and a number,
inequalities can describe a range of values.
expression.
4. The numerical part of a term that contains
a variable is called a sequence.
5. The parts of an algebraic expression
separated by a plus sign are called terms
of sequence.
6. An algebraic expression is in simplest
form if it has no like terms and no
parentheses.
7. In an arithmetic sequence, the difference
between any consecutive terms is the
same.
8. The difference between any two
consecutive terms in an arithmetic
sequence is called the term of expression.
Vocabulary Review at tx.msmath3.com
Chapter 10 Study Guide and Review
563
CH
APTER
10
Study Guide and Review
Lesson-by-Lesson Review
10-1
Simplifying Algebraic Expressions
(pp. 528–533)
Use the Distributive Property to rewrite
each expression.
9. 4(a + 3)
10. (n - 5)(-7)
Simplify each expression.
11. p + 6p
12. 6b - 3 + 7b + 5
13. SOCCER Pam scored n goals. Leo
Example 1 Use the Distributive
Property to rewrite -8(x - 9).
-8(x - 9)
Write the expression.
= -8[x + (-9)] Rewrite x - 9 as x + (-9)
= -8(x) + (-8)(-9) Distributive Property
= -8x + 72
Simplify.
scored 5 fewer than Pam. Write an
expression in simplest form to
represent this situation.
10-2
Solving Two-Step Equations
(pp. 534–538)
Solve each equation. Check your
solution.
14. 2x + 5 = 17
16.
_c + 2 = 9
5
15. 4 = -3y - 2
17. 39 = a + 6a + 11
18. ZOO Four adults spend $37 for
admission and $3 for parking at the
zoo. Solve the equation 4a + 3 = 40 to
find the cost of admission per person.
10-3
Writing Two-Step Equations
Example 2
Solve 5h + 8 = -12.
5h + 8 = -12
Write the equation.
5h + 8 - 8 = -12 - 8 Subtract 8 from
5h = -20
5h
-20
_
=_
5
5
h = -4
The solution is -4.
Simplify.
Check this solution.
(pp. 539–543)
19. Six more than twice a number is -4.
Example 3 Translate the following
sentence into an equation. Then solve.
20. The quotient of a number and 8, less 2,
6 less than 4 times a number is equal to 10.
Translate each sentence into an equation.
is 5.
6 less than
4 times a number
4n - 6
21. MEDICINE Dr. Miles recommended
that Jerome take 8 tablets on the
first day and then 4 tablets each day
until the prescription was used. The
prescription contained 28 tablets. How
many days will Jerome be taking pills
after the first day? Write an equation
and then solve.
564
each side.
-12 - 8 = -12 + (-8)
or 20
Divide each side by 5.
Chapter 10 Algebra: More Equations and Inequalities
4n - 6 = 10
4n - 6 + 6 = 10 + 6
4n = 16
16
4n
_
=_
4
4
n=4
is
10.
=
10
Write the equation.
Add 6 to each side.
Simplify.
Divide each side by 4.
Simplify.
Mixed Problem Solving
For mixed problem-solving practice,
see page 737.
10-4
Sequences
(pp. 544–548)
State whether each sequence is
arithmetic. Write yes or no. If it is, state
the common difference.
22. 2, 5, 8, 11, 14, … 23. 17, 16, 15, 11, 7, …
24.
_3 , 2, _5 , 3, _7 , …
2
2
2
25. 15, 25, 40, 60, 75, …
Write an expression that can be used to
find the nth term in each sequence. Then
write the next three terms.
26. -7, -4, -1, 2, 5, … 27. 3, 6, 9, 12, 15, …
28. 3, 5, 7, 9, 11, …
29. 1, 4, 7, 10, 13, …
Write the nth term of each arithmetic
sequence.
30. 4, 10, 16, 22, … ; n = 15
Example 4 State whether the sequence
13, 8, 3, -2, -7, … is arithmetic. Write
yes or no. If it is, state the common
difference.
13,
-5
8,
3, -2, -7, -12, -17, -22, …
-5
-5
-5
-5
-5
-5
Yes, the sequence is arithmetic because the
terms have a common difference of -5.
Example 5 Write an expression that
can be used to find the nth term of the
sequence -3, -6, -9, -12, -15, . . . .
Then write the next three terms.
Term Number (n)
1
Term
3
2
3
4
5
-6 -9 -12 -15
31. 5, 2, -1, -4, … ; n = 25
32. SAVINGS Loretta has saved $5. Each
week, she adds $1.50. If she does not
spend any of her saved money, how
much will she have after 6 weeks?
10-5
-3 -3 -3 -3
The terms have a common difference of
-3, so the expression is -3n. The next
three terms are -3(6) or -18, -3(7) or
-21, and -3(8) or -24.
Solving Equations with Variables on Each Side
(pp. 553–556)
Solve each equation. Check your
solution.
Example 6
33. 11x = 20x + 18
34. 4n + 13 = n - 8
35. 7b - 3 = -2b + 24
36. 9 - 2y = 8y - 6
37. GEOGRAPHY The coastline of
California is 46 miles longer than
twice the length of Louisiana’s
coastline. It is also 443 miles longer
than Louisiana’s coastline. Find
the lengths of the coastlines of
California and Louisiana.
Solve -7x + 5 = x - 19.
-7x + 5 = x - 19 Write the equation.
-7x + 7x + 5 = x + 7x - 19 Add 7 to
each side.
5 = 8x - 19
5 + 19 = 8x - 19 + 19 Add 19 to
each side.
24 = 8x
8x
24
_
=_
Divide each side by 8.
8
8
3=x
Simplify.
The solution is 3.
Chapter 10 Study Guide and Review
565
CH
APTER
10
Study Guide and Review
10-6
PSI: Guess and Check
(pp. 557–558)
Solve using the guess and check strategy.
38. FUND-RAISER The Science Club sold
candy bars and pretzels to raise money.
They raised a total of $62.75. If they
made $0.25 on each candy bar and
$0.30 on each pretzel, how many of
each did they sell?
39. FOOD A store sells apples in 2-pound
bags and oranges in 5-pound bags.
How many bags of each should you
buy if you need exactly 11 pounds of
apples and oranges?
40. BONES Each hand in the human body
has 27 bones. There are 6 more bones
in the fingers than in the wrist. There
are 3 fewer bones in the palm than in
the wrist. How many bones are in each
part of the hand?
10-7
Inequalities
Example 7 The product of two
consecutive even integers is 1,088.
What are the integers?
The product is close to 1,000.
Make a guess. Try 24 and 26.
24 × 26 = 624
This product is too low.
Adjust the guess upward. Try 30 and 32.
30 × 32 = 960
This product is still too low.
Adjust the guess upward again. Try 34
and 36.
34 × 36 = 1,224 This product is too high.
Try between 30 and 34. Try 32 and 34.
32 × 34 = 1,088 This is the correct product.
The integers are 32 and 34.
(pp. 559–562)
Write an inequality for each sentence.
41. SPORTS Participants must be at least
12 years old to play.
42. PARTY No more than 15 people at the
party.
Example 8 All movie tickets are
$9 and less. Write an inequality for
this situation.
Let t = the cost of a ticket.
t≤9
For the given value, state whether each
inequality is true or false.
Example 9 Graph the inequality
a < -4 on a number line.
43. 19 - a < 20, a = 18
Place an open circle at -4.
Then draw a line and an arrow to the left.
44. 9 + k > 16, k = 6
Graph the inequality on a number line.
45. t < 2
46. g ≥ 92
47. NUTRITION A food can be labeled low-
fat only if it has no more than 3 grams
of fat per serving. Write an inequality
to describe low-fat foods.
566
Chapter 10 Algebra: More Equations and Inequalities
CH
APTER
Practice Test
10
Use the Distributive Property to rewrite each
expression.
1. -7(x - 10)
2. 8(2y + 5)
Simplify each expression.
Find the nth term of each arithmetic sequence.
16. 3, 6, 9, 12, … ; n = 12
17. 18, 11, 4, -3, … ; n = 10
Solve each equation. Check your solution.
3. 9a - a + 15 - 10a - 6
18. x + 5 = 4x + 26
4. 2x + 17x
19. 3d = 18 - 3d
Solve each equation. Check your solution.
_k - 11 = 5
5. 3n + 18 = 6
6.
7. -23 = 3p + 5 + p
8. 4x - 6 = 5x
9. -3a - 2 = 2a + 3
2
20. -2g + 15 = 45 - 8g
21. MUSICAL Joseph sold tickets to the school
musical. He had 12 bills worth $175 for the
tickets sold. If all the money was in $5 bills,
$10 bills, and $20 bills, how many of each
bill did he have?
10. -2y + 5 = y - 1
11. FUND-RAISER The band buys coupon books
for a one-time fee of $60 plus $5 per book. If
they sell the books for $10 per book, write
and solve an equation to determine how
many books they will need to sell in order
to break even.
22. BANKING First Bank charges $4.50 per
month for a basic checking account plus
$0.15 for each check written. Citizen’s Bank
charges a flat fee of $9. How many checks
would you have to write each month in
order for the cost to be the same at both
banks?
Translate each sentence into an equation.
12. Three more than twice a number is 15.
13. The quotient of a number and 6 plus 3 is 11.
14. The product of a number and 5 less 7 is 18.
15.
23.
TEST PRACTICE The perimeter of the
rectangle is 44 inches.
4x in.
TEST PRACTICE In the sequence below,
which expression can be used to find the
value of the term in the nth position?
Position
Value of Term
1
13
2
8
3
3
4
-2
n
?
A -5n + 18
x 7 in.
What is the area of the rectangle?
F 22 in 2
G 120 in 2
H 392 in 2
J 440 in 2
For Exercises 24 and 25, write an inequality
and then graph the inequality on a number
line.
5n
B _
24. COMPUTERS A recordable DVD can hold at
C 13n
25. GAMES Your score must be over 55,400 to
most 4.7 gigabytes of data.
2
D 8n + 5
Chapter Test at tx.msmath3.com
have the new high score.
Chapter 10 Practice Test
567
APTER
10
Texas
Test Practice
Cumulative, Chapters 1–10
Read each question. Then fill in the
correct answer on the answer document
provided by your teacher or on a sheet
of paper.
1. Which expression can be used to find the
nth term in the following arithmetic
sequence, where n represents a number’s
position in the sequence?
3
4
n
C Dante scored exactly half of the total
number of touchdowns.
3
7
11
15
?
D Dante scored the most touchdowns.
B 4n + 1
D 5n - 2
5. The bar graph shows the results of a survey
2. GRIDDABLE Triangle ABC was dilated to
create triangle DEF. What scale factor was
used to create triangle DEF?
D
&AVORITE3CHOOL3UBJECT
C
F
SIC
-U
H
TH
1 2 3 4 5 6 7 8x
%N
GLI
S
y
-A
2
3
4
5
6
7
8
on the favorite school subject among middle
school students. Which statement best
describes why a person reading the graph
might get an incorrect idea about the
favorite subject of middle school students?
.UMBEROF3TUDENTS
8
7
6
5
4
3
2
1
87654321 O
A
B Orlando scored the most touchdowns.
2
C 4n - 1
B
A Eddie scored the most touchdowns.
1
A 3n
E
108 touchdowns this season. Eddie scored 8
more touchdowns than Dante, and Orlando
scored twice as many goals as Dante. Which
is a reasonable conclusion about the number
of goals scored by the players?
NC
E
Position in
Sequence
Term
4. Orlando, Eddie, and Dante scored a total of
3C
IE
CH
F The intervals on the vertical scale are not
consistent.
G The graph does not include gym class.
3. There are 4 children in the Owens family.
H The vertical scale should show the
number of votes for each subject.
1
Jamie is 1_
times as tall as Kelly, and he
J The title of the graph is misleading.
is 6 inches taller than Olivia. Sammy is
56 inches tall, which is 2 inches taller
than Olivia. Find Jamie’s height.
6. GRIDDABLE The probability that Timothy
2
F 52 inches
H 58 inches
G 56 inches
J 60 inches
568
Chapter 10 Algebra: More Equations and Inequalities
2
. About how many
will make a basket is _
9
baskets would you expect him to make in
his next 90 attempts?
Texas Test Practice at tx.msmath3.com
Get Ready
for the Texas Test
For test-taking strategies and more practice,
see pages TX1–TX23.
7. Let n represent the position of a number
10. The table shows the number of hours
in the following sequence. Which expression
can be used to find any term in the
sequence?
students have volunteered at a community
center over several months. If the students
volunteer 290 during the month of
September, which measure of data will
change the most?
_1 , _2 , 1, _4 , _5 , 2, ...
3 3
3 3
A 3n
n
B _
2
C 2n
n
D _
3
Student Volunteer Hours
Jan Feb Mar Apr May May
145 150 125 165 160 155
Month
Hours
F mean
8. Before the last soccer game of the season,
Tony scored a total of 45 goals. He scored
3 goals in the final game, making his season
average 2 goals per game. To find the total
number of games that Tony played, first
find the sum of 45 and 3, and then —
G median
H mode
J They will all change the same amount.
F add the sum to 2.
G subtract 3 from 145.
Questions 9 and 10 When answering a
test question involving a graphic,
always study the graphic and its labels
carefully. Ask yourself, “ What information
is the graphic telling me?”
H multiply the sum by 2.
J divide the sum by 2.
9. A collector’s coin was worth $2.50 in 1999.
Pre-AP
The table shows the value of the coin over
several years. Based on the information in
the table, what is a reasonable prediction for
the value of the coin in 2008?
Record your answers on a sheet of paper.
Show your work.
11. The table below gives prices for two
different bowling alleys in your area.
Coin’s Value
Year
Value of Coin
1999
$2.50
2000
$2.90
2001
$3.40
2002
$4.00
2003
$4.70
2004
$5.50
A $11.00
C $8.50
B $9.70
D $7.40
Bowling
Alley
X
Y
Shoe
Rental
$2.50
$3.50
Cost per
Game
$4.00
$3.75
a. Write an equation to find the number of
games g for which the total cost to bowl
at each alley would be equal.
b. How many games will you have to bowl
at each alley for the cost to be equal?
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569