You will solve two-step and multi-step equations as well as equations with
variables on both sides. You will also read, write, and graph inequalities.
Finally, you will learn to solve inequalities including multi-step inequalities
using the properties of inequalities.
COMMON
CORE
Lesson Standards for Mathematical Content
11-1 Solving Two-Step Equations
11-2 Solving Multi-Step Equations
11-3 Solving Equations with Variables
on Both Sides
11-4 Inequalities
11-5 Solving Inequalities by Adding
or Subtracting
11-6 Solving Inequalities by Multiplying
or Dividing
11-7 Solving Multi-Step Inequalities
Problem Solving Connections
Performance Task
Assessment Readiness
PREP FOR
Unpacking the Standards
Understanding the standards and the vocabulary terms in the
standards will help you know exactly what you are expected to learn
in this chapter.
What It Means to You
You will solve mUlti-step equations by identifying the operations
involved and undoing them in the opposite order.
EXAMPLE
A new one-year membership at Vista Tennis Center costs $160. A
registration fee of $28 is paid up front, and the rest is paid monthly.
How much do new members pay each month?
Let m represent the monthly cost.
+
28+12m=
-28
=
160
-28
12m
=
132
m.
12m __
12 -
m
12
= 11
New members pay $11 per month for a one-year membership.
What It Means to You
You will combine, multiply, and factor expressions. The Commutative,
Associative, and Distributive Properties will help you simplify expressions.
EXAMPLE
Solve 3(z - 1) + 8 =
14.
3(z -1) + 8 = 14
3z- 3 + 8 = 14
3z+ 5= 14
-5
-5
3z = 9
3z
"3
= "39
Z=
3
Distribute 3 on the
left side.
Combine like terms.
Add -5 to both sides.
Divide both sides by 3.
What It Means to You
You will solve one-step and two-step inequalities and graph their
solution sets. You will be able to describe the solution set in the context
of a problem.
EXAMPLE
The 83 members of the Newman Middle School Band are trying to raise
at least $5,000 to buy new percussion instruments. they have already
raised $850. How much should each student still raise, on average, to
meet the goal?
Let d represent the average amount each student should still raise.
83d + 850 ;:::: 5,000
-850
----83d
-850
----~
;:::: 4,150
~c:t > 4,15~
83 - 83
d ;:::: 50
On average, each band member should raise at least $50.
You will decide whether to use expressions, equations, or inequalities
when solving problems. Your prior understanding of arithmetic will be
combined with your algebraic skills to solve real-life problems.
EXAMPLE
I
Jamal owns twice as many graphic novels as Levi owns. Adding 6 to the
number of graphic novels Jamal owns and then dividing by 7 gives the
number Brooke owns. Brooke owns 30 graphic novels. How many does
Levi own?
Let 9 represent the number of graphic novels Levi owns. Then 2g
2g + 6
represents the number Jamal owns and ~'r-- represents the number
Brooke owns.
29 ; 6 = 30
(7)2g ;~ = (7)30
2g+6=210
2g + 6 - 6 = 210 - 6
2g = 204
2i, = 2~4
9
Levi owns 102 graphic novels.
= 102
Key Vocabulary
Addition Property of Inequality (Propiedad de adici6n de /a desiguafdad) The property that _
states that if you add the same number to both sides of an inequality, the inequality will still be true.
Division Property of Inequality (Propiedad de fa divisi6n de fa desigua/dad) The property
that states that if you divide both sides of an inequality by the same positive number, the inequality will still
be true. If you divide both sides of an inequality by the same negative number and reverse the inequality
symbol, the inequality will still be true.
inequality (desiguafdad) A mathematical sentence that shows the relationship between quantities that
are not equivalent.
Multiplication Property of Inequality (Propiedad de multiplicacion de fa desiguafdad) The
property that states that if you multiply both sides of an inequality by the same positive number, the
inequality will still be true. If you mUltiply both sides of an inequality by the same negative number and
reverse the inequality symbol, the inequality will still be true.
solution set (conjunto soluci6n) The set of values that make a statement true.
Subtraction Property of Inequality (Propiedad de sustracci6n de la desigualdad) The
property that states that if you subtract the same number from both sides of an inequality, the inequality will
still be true.
Name _ __ ---
--~
Class _______ Date----,Jf1i?
Solving Two-Step Equations
Essential question: How do you solve equations that contain two operations?
:Pft~tItE\> Solving Two-Step Equations
Lff<'f_<,~
__<,
Carrie and Freddy collect stamps. Carrie notes that she has twelve less
than five times the number of stamps Freddy has. Carrie has 23 stamps.
let f be the number of stamps that Freddy has.
~ Write an equation that represents Carrie's collection. __~______~______,_<_<_
8; Method 1: Solve the equation by covering up the term with the variable.
5f- 12 = 23
Cover the term containing the variable. Think: "Some number minus 12 equals 23." _-12=23 What number minus 12 equals 23? Now uncover the term.
Think: 5 times some number equals 35.
5 times
equals 35.
5f=
f=
c; Method 2: Solve the equation by undoing the operations. Step l: Make a table. Qperationsin the Equation First, list the
operations in
the equation
according to
the order in
which they
are applied to
the variable.
1. Firstfis
by 5.
2. Then, 12 is
1. First _______
12 to both sides of
the equation.
2. Then ___,,_,__~___,_,
both sides by 5.
Then, starting with
the last operation in
the equation write
the opposite of the
step. Continue
writing the opposite
until every step is
accounted for.
Step 2: Apply the steps in the "to solve" column to solve the equation.
5f- 12 = 23
= 23
5f- 12
-<Sf
<-.
35
=~~~
f=
Freddy has _ _ stamps.
Chapter 11
459 Lesson 1
1a.
In what way are these two methods for solving equations similar?
1b. To solve an equation, you isolate the variable by performing _~____
_" order from the order in which they are applied
operations in the
to the variable in the original equation.
Solving Two-Step Equations that Contain fractions
Use a table to help you solve each equation.
1:'.
22=~+7
Solution
22=!!+7
4
1. First n is
1. First _____.__
on both sides of the
equation.
2. Then, _______
22
15
both sides by ____.
SIDutlon
~= 12
both sides by ____.
2.Then, __________
both sides by ___.
"3
2x
= 12
2x
=36
X=
Solve each equation.
~+10=40 Chapter 11
2b. !-9=4
2
460 2e.
~ =6
Lesson 1
\ Solving Two·Step Equations that Contain Decimals
Mai buys 3 new fish for her aquarium for a total of $9.69. Two of the fish
are guppies, and the other is a white cloud. The white cloud costs $3.19.
What is the cost of each guppy?
A Write an equation that can be used to find g, the cost in dollars of each guppy.
times
total cost
Complete the table to decide how to undo the operations in the equation.
B
...•..
.... Qperatiofls i"c .the Equation
1. First g is
1. First
of the equation.
~
_by _ _.
2. Then - - ­ is
...
...
..
TO Solve
to the product.
/
-~-
from both sides
2. Then
both sides by
t· Use the steps from the "To Solve" column to solve the equation.
2g + 3.19 = 9.69
2g+ 3.19
=9.69
2g=
-2g
- = -6.5
.­
g=
The cost of each guppy is ~ ___..~_._.'
<-,"
0
REfLECT
3a.
How could you check your answer to O?
•. IIl!JH1S1
Four friends equally split the cost of a pizza. Garth pays for his share of the pizza plus a
drink that costs $1.29. He pays a total of $4.78. Write and solve an equation to find c, the
cost in dollars of the entire pizza.
Chapter 11
461
Lesson 1
PRACTICE
Solve each equation.
1. 6x+8=26
3. 3f-12 = 24
2. fl.-4=5
3
--~~-----~-
t
_ 1
5. 41 -2
6. 9m + 2.18 = 4.7
4. 2c-14 = 14
-,--~---~-~--~~~--~~~~--
7. 0.2a-4 = 3.6
8. ~ + 8.4 = 12.8
9. Gabriella ran several laps around a i-mile track. Then she ran 2% miles on a trail. In all, she
ran 4 miles. Write and solve an equation to find I, the number of laps Gabriella ran.
10. Mario bought 3 cans of tennis balls. He paid for part of the purchase using a gift card with
$7.25 on it. He paid for the rest with $4.72 of his own money. Write and solve an equation to
find c, the cost of each can of tennis balls.
Use the information about canoe and kayak rentals to solve each
problem.
11. Alisa and Chelsea paid a total of$31 to rent a canoe. Write
and solve an equation to find h, the number of hours they
rented the canoe.
12. Rick paid $24 to rent a kayak. Adam paid $20 to rent a kayak. How much longer did Rick rent
a kayak than Adam did? Explain how you found your answer.
13. Error Analysis Rochelle solve the equation
4x + 2.8 = 3.5 as shown at right. Describe the error
that Rochelle made. What is the correct solution of
the equation?
Ifx + 2.B = ~5
Ifx + 2.B - 2.B = ~5 + 2.B
Ifx ".3
Ifx ".~
=
If=T
x = 1575
Chapter 11
462 Lesson 1
_ _ Class
Name_.
______ Date _ _ __
11-1
Solve. Check each answer.
1. 7x + 8
= 36
4. 6a - 4 = -2
2. -3Y - 7
=2
5. 5k + 2 = 6
3.4a-13=19
6.9m-14=-8
Solve.
v
7. - - 3 = 5
4
f
10. -7+ - =-1
2
u
5
8. - + 3 = 1
w
11.9+ -
4
=-5
z
9.6+ - = 9
9
e
12. - - 3 =-5
7
i:'
'"
Q.
E
8
en
<:
~
:a:::I
Cl-
1:
:::I
d
13. -8 + - =2
5
8
:;;
.s
u
14.-+3=6
5
15. -
f
-3
+5=8
:c
it:
~
<:
o
.Een
:::I
o
:c
@
16. Two years of local Internet service costs $685, including
the installation fee of $85. What is the monthly fee?
Chapter 11
463
Practice and Problem Solving
Write the correct answer.
1. Last week, Carlie had several rice
cakes and 3 granola bars as snacks.
The snacks contained a total of
800 calories. If each granola bar had
120 calories and each rice cake had
40 calories, how many rice cakes did
she have?
2. Jo eats 2,200 calories per day. She
eats 450 calories at breakfast and
twice as many at lunch. If she eats
three meals with no snacks, which
meal will contain the most calories?
3. Erika is following a 2,200 calorie­
per-day diet. She eats the
recommended 9 servings of breads
and cereals, averaging 120 calories
per serving. She also eats 5 servings
of vegetables. If the rest of her daily
intake is 870 calories, what is the
average number of calories in each
serving of vegetables?
4. Brandon follows a 2,800 calorie­
per-day diet. He has 11 servings of
breads and cereals, which average
140 calories each. Yesterday, he had
a combined 9 servings of fruits and
vegetables, averaging 60 calories
each. How many 180-calorie servings
of meat and milk did he have to
complete his diet?
Choose the letter for the best answer.
The table shows calories burned by a
person performing different activities.
5. Kamisha swims for 0.25 hour. How
many calories does she burn?
A 30 calories
C 1.95 calories
B 195 calories
o
117 calories
Swimming
6. Stu jogs at a rate of 5 mi/h. How far
must he jog to burn 418.5 calories?
F 9 mi
H 3.75 mi
G 4.65 mi
J 45 mi
Jogging
7. Terry rides her bike for 40 minutes
and plays basketball for an hour. How
many calories does she burn?
8. How many hours would you have
to ride your bike at 10 mi/h to burn
550 calories?
A 67 calories
C 670 calories
F 1.67 hr
H 1.0 hr B 560 calories
calories
0 1,300
G 1.5 hr
J 0.75 hr Chapter 11 464
Practice and Problem Solving
_ _"____
Name
Class_~
_ _ __
Solving Multi-Step Equations
Essential question: How do you solve equations that contain mUltiple
operations?
+_=0
Key
Remember
= positive x +.=0
Combining like Terms to Solve Equations
It costs $3 per person to visit a petting zoo, $1 per person to feed the
animals, and $2 for parking. A family paid $14 to visit the zoo, feed the
animals, and park their car. The equation 3x + x + 2 = 14 can be used to
find x, the number of people in the family. How many people are in the
family?
A Use algebra tiles to model the equation 3x + x
+ 2 = 14. The drawing shows tiles representing the left side of the equation. Draw the tiles representing the right side of the equation. B Isolate the positive x tiles on the left side of the model.
To do so, remove the same number of + 1 tiles from each side of your model.
How many + 1 tiles did you remove from each side? ___
There are now ___ positive x tiles on the left side and _~ __ + 1 tiles on
the right side.
C To find the value of x, divide each side ofyour model into 4 equal groups.
There is
positive x tile in each group on the left side and _ _ "_, + 1
tiles in each group on the right side.
So, the solution of the equation is x = _"""___.
There are __~_ people in the family.
TlIYTHIS!
Use algebra tiles to solve each equation.
1a.
4x + 1 + 2x = 13
Chapter 11
1 b.
5x - 3x + 3 = 11
465 1c.2x+x-2=1
Lesson 2
1d.
How could you have solved the equation in 0 without using algebra tiles?
~,:Jlill'f[,~;ISi~\ Solving Equations Using the Distributive Property
Kara used the formula P = 2(t + w) to find the perimeter of a
photograph. She tells Jim that the length is 6 centimeters and the
perimeter is 22 centimeters. How can Jim find the width of the photo?
:'A,, Rewrite the formula, substituting the values that you know.
•.!i.' Method 1
= 2(6 + w)
-~--~--.~
Since (6 + w) is being _ _ _ _~
by ___,
by 2 on both sides of the equation.
6+w
Simplify.
11=
6+w
Then,
=
w
Simplify.
+
w
from both sides.
Method 2
22 =
(6)
22 =
+ 2w
22 =
12 + 2w
=
.!!2 =
2;
2w
Use the Distributive Property.
Distribute __ to each term in parentheses.
Simplify.
Then, _ _ __
_ __ from both sides.
Simplify.
2w
2
Simplify.
Solve each equation.
2a.
10 = 4(3 + x)
Chapter 11
2b. 40(x - 2) = 200
466
2e.
~(2x + lO) = 35
Lesson 2
2d. How are the two solution methods alike?
How are the two solution methods different?
..
~:··i .aIG~
a
. . ~. }~E~~~~p'~~':\ Solving Problems Using Equations
60~inches Carl is hanging a picture, and he wants
to center it on the wall. The picture is 18~ inches long, and the wall is 60~ inches long.
A
x inches
18~inches
x inches
Estimate how many inches from each side of the wall the picture should
be placed.
B
Use the diagram to write an equation to find the exact distance from each side the picture
needs to be placed.
+
=
Total length
of wall
Distance from
side of wall
+
Distance from
side of wall Total length
of picture
Combine like terms and solve the equation for the variable.
60~4
=
=
+
-~-2-
=
C
The picture should be placed _~_______ from each side of the wall.
REFLECT
3a. Was your estimate reasonable? Explain your answer.
3b. What If... ? If the picture were 24 inches wide, how would the amount of space on either
side of the wall change?
Chapter 11
467 Lesson 2
PRACTICE '
Solve each equation.
1. 4x-2x+ 3 = 7
2. 8 = 3n + 2n - 7
5. 0.3(b - 21) = 3.6
6. 30 = 12 + 6a - 3a
3. 5(c + 4) = 25
4.
!(t + 6) = 27
8. 4.2v + 1.8v = 54
9. Ms. Pryce has 82 books. She puts 10 of them in her desk. The rest will go in her bookcases.
One bookcase has 2 shelves, and the other has 4 shelves. Write and solve an equation to find
s, the number of books Ms. Pryce should put on each shelf so that each has the same
number of books.
10. Tubes of oil paint are on sale for $0.50 off. Cody bought 6 tubes of paint on sale and paid a
total of $25.80. Write and solve an equation to find r, the regular price in dollars of each tube
of oil paint.
The photo shown is framed in a blue border.
Use the dimensions in the diagram for each
problem.
11. Write and solve an equation to find x, the left
and right width of the border.
12. Write and solve an equation to find y, the top
and bottom width of the border.
13. Estimation Explain how you can use estimation to check that your answer to problem
12 is reasonable.
Chapter 11
468 lesson 2
11-2
Name _ _ _ ~ _ _ ~ _ _"_______ Class
Additional Practice
7
Solve.
1. 15x - 8 - 3x = 16
2. 5n + 3 + 4n = 30
3. h-6+7h==42
4. -3g+6+2g= 15
5. -2b + 7 - 3b = 2
6. 5y+ 1 + 3y== -15
7.4k-14+3k==21
8. 9m + 10 - 14m ==-5
9. -2d+ 18-4d==60
---~"--
10. 3(n + 5) + 2 = 26
13. 2.4(m
3) + 3.8 == -8.2
11. 4 - 2(v - 6) ==-8
3
14. 6==8(s- 4)-20
12. 1.4 - 1.6(t + 6) == 4.6
4
5
15. 5( c + -) + 6 == 50
16. Joel has twice as many CDs as Mariella has. Subtracting 7 from
the number of CDs Joel has and dividing by 3 equals the
number of CDs Blake has. If Blake has 25 CDs, how many CDs
does Mariella have?
Chapter 11
469
Practice and Problem Solving
Write the correct answer,
To convert a temperature from degrees Fahrenheit to degrees
Celsius, you can use the formula CF - 32)0.56 = °C.
1. The record high temperature in North
Carolina is 110 of. What is the record
high in degrees Celsius?
2. The record low temperature in
Florida is -2 of. What is the record
low in degrees Celsius?
3. The record high temperature in the
United States is 134 of. This was
recorded in Greenland Ranch,
California, on July 10,1913. What is
that temperature in degrees Celsius?
4. The record high in Texas is 120 OF.
The record low in Texas is -23 of. In
degrees Celsius, what is the range
between the record high and low
temperatures in Texas?
5. When the temperature is 4°C, you
need to wear a heavy coat. Write 4 °C
in degrees Fahrenheit.
6. When the temperature is 28°C. you
might want to go to the beach. Write
28 °C in degrees Fahrenheit.
Choose the letter for the bestanswer.
8. Three friends each pay $4.15 to buy
a pizza. A basic pizza costs $9.45.
Additional toppings cost $1 each.
How many toppings were on the
pizza?
7. Faith spent $78 at Fashion
Warehouse. She bought 2 shirts that
each cost $17.50 and a pair of
shoes. How much did she spend for
the shoes?
A $34.00
C $60.50 F 2 toppings
H 4 toppings S $43.00
0 $113.00 G 3 toppings
J 5 toppings 9. Todd buys 3 CDs at $16.99 each and
a OVO that costs $24.99. He pays
with a $100 bill. How much change
does he receive?
10. Marina bought 4 books. Jose bought
half as many books as Sen bought.
Together. the 3 friends bought 13
books. How many books did Sen buy?
A $24.04
C $49.03 F 9 books
H 3 books S· $8.04
o
G 6 books
J 2 books Chapter 11 $67.96 470
Practice and Problem Solving
Class ________~
Name _ _ _.
Solving Equations with Variables
on Both Sides
Essential question: How do you solve equations that contain variables
on both sides?
Modeling Equations with Variables on Both Sides
Jessie and Kenya are buying supplies for their scrapbooks. Jessie buys 4
sticker packs. Kenya buys 2 sticker packs plus a $4 pair of scissors. Each girl
pays the same amount. The equation 4x = 2x + 4 models this situation,
where x is the cost of each sticker pack. What is the cost of each sticker pack?
A
Use algebra tiles to model the equation 4x =: 2x + 4.
The draiNing shows tiles representing
the left side of the equation. Draw the
tiles representing the right side of
the equation.
B
Get all x tiles on one side of the model.
Remove the positive x tiles from the right side ofyour modeL To keep
the equation balanced, remove
positive x tiles from the left
side of your modeL
There are now
positive x tiles on the left side and
______ + 1 tiles on the right side.
C
To find the value of x, divide each side ofyour model into 2 equal groups.
There is
positive x tile in each group on the left side and
____ + 1 tiles in each group on the right side.
So, the solution of the equation is x =:
Each sticker pack costs $ _______ .
_____,
TRY THIS!
Use algebra tiles to model and solve each equation.
1a.
3x + 3
= 2x + 8
1b. 6x - 6
= 3x -
3
1c. x
+ 7 = 5x -
1
REFLECT
1d.
How could you have solved the equation in 0 without using algebra tiles?
Chapter 11
471
Lesson 3
PRACTICE
Solve each equation.
= 5n
1. 4x + 6 = 3x + 9
2. -3n + 8
4. 24m + 64 = 40m
5. lAg + 204 = 0.8g + 8.1
3. 8d + 9 = 10d - 13
6.
iz=tz+ 12
7. An artist can finish 3 small paintings in the same time it takes her to finish
1 large painting and 1 small painting. It takes her 7 hours to finish a large
painting. Write and solve an equation to find s, the number of hours it takes
the artist to finish a small painting.
8. At a sandwich shop, 6 sandwiches and 2 drinks cost the same as 5 sandwiches and 5 drinks. Each sandwich costs $4.50. Write and solve an equation to find d, the cost in dollars of each drink. 9. Explain how you can check your answer to problem 8.
Use the information about the DVD rental p.lans to solve each problem.
10. Write and solve an equation to find m, the number of months
for which the total cost of the two plans is the same.
11. What If... ? What if Zamex reduced its one-time fee to $8? What
would be the number of months for which the total cost of the
two plans would be the same?
12. Javier uses algebra tiles to model and solve the equation 2x + 3 = x. He starts by removing
one positive x tile from each side. What tiles remain on each side of the model? What should
Javier do next to solve the equation using the model?
--Chapter 11
472
--------------._----_._--­
Lesson 3
Name _ __
11-3
_ _ _ _ _ _~_ _ _ Class _ _ _ _ _, Date _ _ __
Additional Practice) Group the terms with the variables on one side of the equal
sign and simplify. Do not solve.
1. 1ot = 6t + 24
2. -6x - 32 = 2x
3. j = 20 - 4j
4. -5d+40=5d
5.9m-28=2m
6.-x=8+-x
8. 32 - 5v = 3v + 8
9. -12y-10=-6y+14
8
4
9
9
Solve.
7. 8k= 6k- 26
5
3
10.-a+6=-a
8
4
11.
1
2
n+ 10=-n
4
3
12.20-
1
3
d=-d+16
5
10
13. Members of the Lake Shawnee Club pay $40 per summer
season plus $7.50 each time they rent a boat. Nonmembers pay
$12.50 each time they rent a boat. How many times would both a
member and a nonmember have to rent a boat in order to pay
the same amount? - - -
Chapter 11
473
Practice and Problem Solving
\
J
Write the correct answer.
1. Five added to twice Erik's age is the
same as 3 times his age minus 2.
How old is Erik?
2. Three times the perimeter of a
triangle is the same as 75 decreased
by twice the perimeter. What is the
perimeter of the triangle?
3. The area of a pentagon increased by
27 is the same as four times the area
of the pentagon, minus 15. What is
the area of the pentagon?
4. To repair body damage on a car,
AutoBody charges $125, plus
$18 per hour. CarCare charges
$200, plus $12 per hour. Determine
the number of hours for which the
two body shops will cost the same.
Choose the letter for the best answer.
5. Sandy and Suzanne are planting
flower pots around the school
building. Sandy has planted 33 pots
and is planting at the rate of 10 pots
per hour. Suzanne has planted
25 pots and is planting at the rate of
14 pots per hour. In how many hours
will they have planted the same
number of flower pots?
A 3 hr
C 2 hr
B 2.5 hr
D 1 hr
6. The length of the sides of a square
measure 2x - 5. The length of a
rectangle measures 2x, and the
width measures x + 2. For what
value of x is the perimeter of the
square the same as the perimeter
of the rectangle?
C 20 mi B 21 mi
D 15 mi Chapter 11 H x=10 G x=7
J x=12 8. Toni bought some beach towels on
sale for $8 each. Theo bought the
same number of beach towels at the
full price of $12. Toni's total was $24
less than Theo's total. How many
beach towels did they each buy?
7. Louisa used Downtown Taxi, which
charges $2 for the first mile and
$1 .10 for each additional mile. Pietro
used Uptown Cab, which charges $5
for the first mile and $0.95 for each
additional mile. They paid the same
amount and traveled the same
distance. How far did they travel?
A 25 mi
F x=2
474
F 6 towels
H 9 towels G 8 towels
J 12 towels Practice and Problem Solving
Name ________~
~
_ _ _ _ _ _ Class _ _ __
Inequalities
Essential question: How do you read, and write inequalities?
An inequality is a mathematical statement that two quantities
are not equal or may not be equal. The solution set of an
inequality with a variable consists of all possible solutions of
the inequality.
Writing an Inequality
Write an inequality to represent each situation. Then describe the
solution set of the inequality.
A
There are fewer than 6 tigers living in a forest.
Could the number of tigers be...
Yes/No
.. .Iess than 6?
...equal to 6?
...greater than 6?
Let t represent the number of tigers in the forest. Write the correct inequality
symbol to complete the inequality: t
6.
Think about the solution set of the inequality in this situation. Circle the
numbers in the list below that could represent the number of tigers.
>,
c::
-2
5.5
1
'"
4
'"
Because t is a number of tigers, the value of t is limited to whole numbers /
integers / rational numbers less than 6.
0.
E
0
u
c::
~
li
1
3
2
:>
0­
t:
So, an inequality that describes the situation is _____ where t is a
_______~__ that represents the number of tigers.
:>
J
0
~
'"c::
::t:
!E
~
B
You can only take fish from the lake that are at least 14 inches long.
c::
B
.c::
'"
:>
0
::t:
@
Could the length of a fish taken from
the lake be...
Yes/No
... Iess than 14 inches?
...equal to 14 inches?
...greater than 14 inches?
Let/represent the length, in inches, of a fish that can be taken from the lake.
14.
Write the correct inequality symbol to complete the inequality: /
Chapter 11
475
Lesson 4
Think about the solution set of the inequality in this situation. Circle the
numbers in the list below that could represent the length in inches.
15
16.5
20l
22
Is-!
14
Because fis a length in inches, the value off can be any rational number
greater than or equal to 14.
So, an inequality that describes the situation is
, where fis a
~_______ that represents the length of a fish in inches.
1a. What keywords in
and' helped you choose the correct
inequality symbol?
1b. List all of the numbers in the solution set of the inequality in;c~ .. Can you
list all of the numbers in the solution set of the inequality in,.? Explain.
Writing a Situation for an Inequality
Give an example of a real-world situation that can be modeled by the
inequality a > 3, where a is a whole number.
'Il.. First, decide what the variable represents.
Let a represent the weight, in ounces, of an apple.
The possible values of a can only be
les'm a b ago
whole numbers. Underline the definition Let a represent th e numb er 0 f
app
of a that best meets this description.
:. Interpret the inequality symbol.
... fewer than 3?
... exactly 3?
... more than 3?
So, the inequality a > 3 can be modeled by this situation:
2a.
Give an example of a real-world situation that can be modeled by the
inequality m ~ 120, where m is a positive number.
.
Chapter 11
476
Lesson 4
.....-/.
Solutions of Equations and Inequalities
Follow these steps to explore the relationship between the solution
of an equation and the solution set of a related inequality.
A
For each row in the table below, solve the equation in the first column,
and write the solution of the equation in the second column.
For example, the first equation is x + 2 = 4. The value of x that makes this
equation true is 2, so the solution of the equation is x = 2.
B
Next, look at the related inequality in the third column, and circle the values of x in the fourth column that make the inequality true. For example, the inequality x + 2 < 4 is true when x is 0 or I, but it is
not true when x is a whole number greater than 1.
x+2<4
x+2<4
?
x+2<4
?
?
0+2<:4 1+2<:4
2+2<:4
2<4 3<4
4<:4
?
o is a solution.
1 is
a solution.
2 is not a solution.
C In the last column, write the solution set of the inequality.
For example, you found that the inequality x + 2 < 4 is true only when x
is less than 2, so the solution set of the inequality is x < 2.
Equation
Solution of
Equation
Related
Inequality
Which values of x make
the inequality true?
Solution Set
of Inequality
x+2=4
x=2
x+2<4
~2,3,4, 5,6,7,8,9
x<2
x+1=6
X=
x+1<6
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
x<
x-3=0
X=
x-3>0
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
x>
c:
x+1=5
X=
x+1::::;5
0, 1,2,3,4, 5,6, 7, 8, 9
x::::;
::0
2x=4
X=
2x;;:: 4
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
x;;::
'"c:
"'
E
Q.
Q
U
01
~
:l
0­
1::
:l
Q
::
"'
:r:
oS
:E
~
REflECT
Za. Compare the equations and the related inequalities in the table. What is the
only difference between them?
c:
Q
1:
01
:l
o
::c
@
2b.
The solution set of an inequality is x < 4. Is x = 4 a solution of the inequality?
What ifthe solution set is x S 4? Is x = 4 a solution? Explain.
Chapter 11
477 lesson 4
PRACTICE Write an inequality to represent each situation. Then describe the
solution set of the inequality.
1. At most, 15 students can be in the computer lab.
2. For the next hour, the plane's altitude remained above 30,000 feet.
3. The temperature of the ice in an ice rink should be no more than 26 oF.
----------------~--------
-----
Give an example of a real-world situation that can be modeled by each
inequality.
4. f"2: 20, where fis a whole number -----.--­
5. w < 8, where w is a positive number _ _ __
Solve each equation. Then find the solution set of the related inequality.
6. s+2=6
s= - -
8. 3d=9
d= --~-
7. n-l=5
s+2"2:6
s"2: ___
n= - - -
3d> 9
9. t+ 4=4
d> ___
t=
n-l<5
n<
t+4:$4
t:$
10. Critical Thinking The inequality d < 5 represents the distance d, in miles,
that Janelle walks each day. Is the inequality d < 5 true when d = -11 Explain
whether a solution of d = -1 makes sense in this situation.
11. Error Analysis Miguel wrote the inequality t:$ 350 to represent the situation
"The oven temperature must be no less than 350°F." Describe the mistake
that Miguel made. What is an inequality that correctly describes the situation?
Chapter 11
478 Lesson 4
!
;ti
11-4
:~~;z,;jt:g~'Pt~::!:(tSpr:,~'!i~?'i)f0::{!Tif"
Additional practice}
,j" ·,5,
Write an inequality to represent each situation. Then describe
the solution set of the inequality.
1. The temperature today will be at most 50 OF.
2. The temperature tomorrow will be above 70 OF.
3. There was fewer than 20 items in a bag.
4. There must be at least 3 people on the ferry for it to cross the river.
Giv~ an example of a real·world situation that can be modeled
by each inequality.
5. t s -2, where t is an integer
6. j> 5, where j is a whole number
------7. Y sO, where y is a negative number
8. b <
~, where b is a rational number
Chapter 11
479
Practice and Problem Solving
Solve each equation. Then find the solution set of the
related inequality.
j - 7 = 21
1. .
J= 2.
3y=66
y= f +40= 72
3. f=
p
-=-12
4. 4
p= j - 7 ~ 21
j~ ____
3y<66
y<
f +40 :s72
f:s
p
->-12
4
p>
Write an inequality to represent each situation. Then describe
the solution set of the inequality.
5. At least 25 people must be in line for the doors to open for the concert. 6. The temperature of the oven should be no more than 450 of
when baking the dessert.
Chapter 11 480
Practice and Problem Solving
Name_~~~
. ________._.__.____ Class ___ ~___~______
Solving Inequalities by Adding or
Subtracting
Essential question: How do you solve inequalities that involve one operation?
Addition Property of Inequality: You can add the
same number to both sides of an inequality, and
the inequality will still be true.
Example: 2 < 4
Subtraction Property of Inequality: You can
subtract the same number from both sides of an
inequality, and the inequality will still be true.
Example: 2
2+1<4+1
3<StI
<4
2-1 <4-1
1<3t1
Solving Inequalities by Adding
Kate took $3 out of her purse, and she still had at least $8 in it. How
much did she have to begin?
The phrases at least or at most can be confusing. At least means that amount or
more, so use the greater than or equal to (;:::) symboL At most means that amount
or less, so use the less than or equal to symbol (s).
A
Write an inequality to represent the amount of money in Kate's purse.
B
Use inverse operations to solve the inequality.
m - 3 ;::: 8
m-3;:::
to each side.
Simplify.
8
When graphing an inequality on a number line, use a solid circle to show that the variable can
equal that value. Use an empty circle to show that the variable cannot be equal to that value.
Since money is not just integer values, you can shade a solid arrow, or ray, to the right
C
Graph the solutions on a number line.
0(
I
o
I
1
2
3
4
5
6
i
7
8
9
I.
10 11 12 13 14 15 16 17 18 19 20
o What does the solution tell you?
Chapter 11
481
Lesson 5
Solve. Then graph the solution.
1a.
x
+4<9
•I
I ..
1
1 b.
234
5
6
7
Choose a valUEi in the shaded area of the number line from
Substitute it into the original
inequality from
Does this value make the inequality true?
Now choose ~ value outside the shaded area of the number line from
the original inequality. Does this value make the inequality in ' A, true?
Substitute it in
I
Conjecture What does the shaded part of the inequality show?
Solving Inequalities by Subtracting
A suitcase must:weigh less than 50 pounds, or an extra fee will be
charged. Patrick~s suitcase now weighs 38 pounds. How much weight
could Patrick add to the suitcase without paying an extra fee?
Write an inequality to represent the situation.
Let a repre:;;ent the weight, in pounds, Patrick adds to the suitcase.
Think:
Use inverse operations to solve the inequality.
38
+ a < 50
Think: 38 is added to the variable, so
~____ 38 from each side to
undo the addition.
a<
Graph the solution set of the inequality.
Think: a represents the additional weight Patrick adds to his suitcase,
so a camiot be negative.
To graph the solution set, draw an empty / a solid circle at 12, and then
shade all positive numbers to the left / right of 12.
«I
-4 -2
Chapter 11
I ..
0
2
4
6
8
482
10 12
14 16
18
Lesson 5
D
Interpret the solution set. What does the solution set mean in the context of the situation? REFLECT 2a.
Explain how you could check that you found the solution set correctly. 2b. What If...? What if the souvenirs Patrick buys on his trip weigh 14 pounds?
Will he have to pay an extra fee ifhe packs them in his suitcase without
taking anything else out? Explain.
TRY THIS! 2e.
Solve the inequality 25
+ x > 32. Then graph the solution set. (I
-10 -8
I•
-6
-4
-2
0
2
4
6
8
10
Solving Inequalities
Allison solved the inequality n
as shown at the right.
+ 3.4 :s; 5.8
f}
f}
3a.
+ ,fT
~
5.8
- 3fT - 5.8
s---O
Identify the error that Allison made. Explain your reasoning.
-------------------------------------
3b. What is the correct solution set of the inequality? __~_____.___.__
3e,
How could Allison have determined that her answer is incorrect?
Chapter 11
483 lesson 5
PRACTICE Solve each inequality, and graph the solution set. 1. x+8~4
-~~--~--
•I
-5 -4
-3
2.
-2
-1
0
2
3
4
I "
5
3. k+.!>Z
4 8
4
-"4
I
-1
3
1
1
-"4 -2 -"4
0
1
1
"4 2
3
"4
I•
5
"4
-~----.-
•I
-50 -40 -30 -20 -10
4. r-0.2
I
5
:~
t-15<30
4
~
20
30
I•
0
10
40
50
0
0.2 0.4 0.6 0.8
1
0.6
I
-1 -0.8 -0.6 -0.4 -0.2
I•
Solve each inequality, and explain what the solution set means in the
context of the situation.
5. At most, 47 passengers can sit on a bus. There are already 29 passengers seated on the bus.
The inequality p + 29 ~ 47 represents this situation, where p is the number of additional
passengers who could be seated.
6. The temperature is currently above 0 °C, but it has fallen 8 °C in the past 6 hours. The
inequality t - 8 > 0 represents this situation, where t is the temperature, in degrees Celsius,
6 hours ago.
The bar graph shows the number of cans students have collected during
a canned food drive. Use the graph for each problem~
7. The 7th grade's goal is to collect at least
160 cans. Write and solve an inequality
to find how many more cans the 7th grade
must collect to meet their goal.
8. Write an solve an inequality to find how many
more cans the 6th grade must collect to beat
the 8th grade's cUrrent number of cans.
o
9. Critical Thinking Is 21 in the solution set of the inequaUtyyou wrote in problem 81 Explain
your answer in the context of the situation.
- - - -..
Chapter 11
484 -.~--.-----
lesson 5
Name_~~
11-5
__
ditional Practice Solve. Then graph each solution set on a number line.
3. x + 4 < -1 ~___________
I ..
< 1
4. h+20 > 2 _ _ _ _.
«
6.s
I
I•
7<-16 _ __
Solve. Check each answer.
7. 41 + g> 27 10. z + 27 < 16
8. w + 23
11. -3
~
~
-18
t + 17 9. a + 15 ~ 9
12. 78
~
b + 64
13. In order for a field trip to be scheduled, at least 30 students
must sign up. So far, 23 students have signed up. At least
how many more students must sign up in order for the field
trip to be scheduled?
-~
Chapter 11 ._------_._------_._---_._---------_._---­
485
Practice and Problem Solving
Write the correct answer.
1. A small car averages up to 29 more
miles per gallon of gas than an SUV.
If a small car averages 44 miles per
gallon, what is the average miles per
gallon for an SUV?
2. Carlos is taking a car trip that is
more than 240 miles, depending on
the route he chooses. He has already
driven 135 miles. How much farther
does he have to go?
3. Driving into the city usually takes
25 minutes. If there is a lot of traffic,
the trip can take up to 45 minutes.
How much additional time should you
allow during a heavy traffic period?
4. To qualify for the heavyweight
wrestling division, Kobe must weigh
at least 180 pounds. If Kobe weighs
168 pounds now, how much weight
should he gain?
Choose the letter for the best answer.
5. On one day, the range of
temperatures in one state was at
most 2r. If the lowest temperature in
the state was 59°, what was the
highest temperature?
At> 86°
B
t ~ 86°
C t = 86° D
6. The highest possible score on the
Scholastic Aptitude Test is 2,400.
Rebecca scored 1,780. She needs a
score of at least 1,950 to qualify for a
scholarship. How much higher must
her score be?
G
7. Romero is saving to buy an Apex
Model 12 Computer. The lowest price
that Romero can find for the
computer is $1,250. Romero now has
$825. His grandmother is going to
give him another $200. How much
more money does Romero need?
A x < $225
C x;::: $225 B x< $425
D x~ $425 Chapter 11 ~
170
H s
S~
450
J
F s
t ~ 86° ~
170 s ~ 450 8. The seating capacity of the school
gym is 550. So far, there are 210 fans
at a basketball game. How many
more fans could attend the game?
F f~ 340
G f> 210
f~
340
J f< 340
H
486
Practice and Problem Solving
c
11-6
Solving Inequalities by Multiplying or
II
rii
Dividing
[!]
Essential question: How do you solve inequalities that involve mUltiplying or
dividing?
A
:
Complete the tables.
Inequality
Multiply each
side by:
3<4
2
r"'~3
-1 S 6
New Inequality
New Inequality is
True or False?
New Inequality
New Inequality is
True or False?
3
5
-
5> 2
-1
1S 7
-5
-8> -10
-8
Inequality
Divide each
side by:
4<8
4
12
~
15
3
-16 S 12
-4
15> 5
L
I
L
-5
B
When both sides of an inequality are multiplied or divided by a
inequality is no longer true.
C
Complete the tables.
Inequality
Multiply
each side by:
New
Inequality
5>2
-1
-5> -2
1S 7
-5
-5 S -35
-8> -10
-8
64> 80
Chapter 11
487 _~_~ ___ number,
Reverse the
Inequality Symbol
the
Reversed .symbol
makes it
True or False?
Lesson 6
.
15> 5
-5
-3> -1
1. Conjecture When both sides of an inequality are multiplied or divided by a negative
number,you
to make the statement true.
• If you multiply both sides of an inequality by the same positive number,
the inequality will still be true.
• If you multiply both sides of an inequality by the same negative number
and reverse the inequality symbol, the inequality will still be true.
• If you divide both sides of an inequality by the same positive number, the
inequality will still be true.
• If you divide both sides of an inequality by the same negative number and
reverse the inequality symbol, the inequality will still be true.
Solving Inequalities by Multiplying
Five friends plan to share the cost of a meal at a restaurant. Each friend's
share is at most $13. The inequality ~ :s; 13 can be used to find c, the cost
in dollars of the meal. What is the solution set of the inequality, and
what does it represent in this situation?
Use inverse operations to solve the inequality.
·13
Think: The variable is divided by 5, so
each side by 5 to undo the
multiplication. The inequality symbol does I
does not need to be reversed.
Interpret the solution set.
What does the solution set tell you? Does it make sense for the cost of the meal
to be $0 or less than SO? Explain.
Chapter 11
488 Lesson 6
C Graph the solution set of the inequality.
(I
-20 -10
I•
0
10
20
30
40
50
60
70
80
TRY THIS!
2a.
Solve the inequality -14x 2:: 42. Then graph the solution set.
(I
-5 -4
-3
-2
-1
0
2
3
4
I•
5
Solving Inequalities by Dividing
A scuba diver's elevation compared to sea level is changing at a constant
rate of -40 feet per minute. He started at the water's surface, and his
elevation is now less than -100 feet. How long has the diver been
descending?
A
Write an inequality to represent the situation.
Let t represent the time, in minutes, the diver has been descending.
Think:
the rate of eleva­
tion change
times
the time
in minutes
the given
etevation
is less than
-100
B
Use inverse operations to solve the inequality.
-40t < -100
-40t
-100
Think: The variable is multiplied by -40, so
__~_~_____ each side by -40 to undo the
multiplication. The inequality symbol does I
does not need to be reversed.
t
c Interpret the solution set.
D
Graph the solution set of the ineqUality.
.,..+1-t--t---'I---+--+--t--t--+--t--t--+-I•
-5
-4
-3
-2
-1
0
2
3
4
5
REFLECT
3. Explain why you did or did not need to reverse the inequality symbol when solving
the inequality.
Chapter 11
489 Lesson 6
PRACTICE Solve each inequality, and graph the solution set.
1. :!<l
3-4
2. JL>3
-6
----~~~--
•I
5
-4
-1
3
1
1
-4 -2 -4
0
1
4
1
2
4
4
4.
I
-10 --8
-6
-4
-2
0
2
4
6
8
I
-25 -20 -15 -10 -5
5
3. O.4s < 3.6
E
E
I•
~
I•
10
0
5
I•
10
15
20
25
i
I'
2
3
4
5
12p:$ -48
<E
I
-5
-4
-3
-2
-1
0
Solve each inequality, and explain what the solution set means in the
context of the situation.
5. Sandra has more than 90 baseball cards. She keeps the cards in 6 boxes,
with the same number in each box. The inequality 6n > 90 represents this
situation, where n is the number of cards in each box.
6. The side length of a square mirror should be no more than 50 inches. The
inequality ~:$ 50 can be used to find P, the perimeter in inches of the mirror.
Tyrone has a gift card to an online entertainment
store with $21 left on it. The chart shows the
store's prices. Use this information for each problem.
7. Write and solve an inequality to find how many
movies Tyrone could buy by using the card.
Song (buy)
$1.20
Music video (buy)
$2.00
Movie (rent)
$3.50
Movie (buy)
$12.50
8. Tyrone uses his card to buy 8 items of the same type. Which of the items in
the table could he have bought? Use an inequality to explain how you
found your answer.
Chapter 11
490 Lesson 6
Additional Practice
11-6 7
Solve.
1.
!2
4.
-~·<-7 S
5
1.6
t
-6 b
8
3. - 2':-9
3
2 . -3 >-B r
s
5. - - s-5
-12
6. S6
5.3
B. -1682':-24
9. -12t>9
Solve. Check each answer.
7. Be < -64 10. -3s S -1BO
11. 1Bb > -24
12. -6m 2': 4
13. It cost Sophia $530 to make wind chimes. How many wind
chimes must she sell at $12 apiece to make a profit?
14. It cost the Wilson children $55 to make lemonade. How many
glasses must they sell at 75¢ each to make a profit?
15. Jorge's soccer team is having its annual fund raiser. The team
hopes to earn at least three times as much as it did last year.
Last year the team earned $B7. What is the team's goal for
this year?
Chapter 11 491
Practice and Problem Solving
Write the correct answer.
1. U.S. Postal Service regulations state
that a package can be mailed using
Parcel Post rates if it weighs no more
than 70 pounds. What is the
maximum number of books weighing
12 pounds each that can be mailed
in one box using Parcel Post rates?
2. Marc wants to buy a set of at least
6 antique chairs for his dining room.
He has decided to spend no more
than $390. What is the most he can
spend per chair?
3. Alfonso earns $9.00 per hour
working part-time as a lab technician.
He wants to earn more than $144
this week. At least how many hours
does Alfonso have to work?
4. Mrs. Menendez invited 8 children to
her son's birthday party. She wants to
make sure that each child gets at
least 4 small prizes. At least how
many prizes should she buy?
Choose the letter for the best answer.
5. The Computer Club spent $2,565 on
mouse pads. The members plan to
sell the mouse pads during the book
fair. If they charge $9.50 for each
mouse pad, how many must they sell
in order to make a profit?
~
A n 5271
C n
B n 5 270
D n ~ 270 F
271 C c ~ 91 B c 590
Des; 91 Chapter 11 p~$1.25
G P ~ $1.50
7. In 2000, the national ratio of students
to computers with Internet access in
public schools was 7:1. Winston
School had 623 students. If the
school had a lower ratio, how many
computers with Internet access did
Winston School have?
A c~ 90
6. The Parents Organization bought
1,000 bumper stickers at $1.25 each
to sell at football games. They want
to make at least $750 profit. What
should be the selling price of the
bumper stickers?
492
H
p~$1.75 J P ~ $2.00 8. A new theme park averaged fewer
than 2,000 visitors per week during
the winter months. What was the
average daily attendance?
~
290
H a > 385 Gas; 186
J a < 286 F a
Practice and Problem Solving
Name __
Solving Multi-Step Inequalities
Essential question: How do you solve multi-step inequalities?
Solving Two-Step Inequalities
As a salesperson, you are paid $52 per week plus $3 per sale. This week
you want your pay to be at least $100. Write an inequality for the
number of sales you need to make, and describe the solutions.
A
Write an inequality to represent the number of sales you need in order to be paid at least $100 for the week. B
Method 1: Solve the inequality by covering up the term with the variable.
3x + 52 2:: 100
e+
Cover the term containing the variable.
Think: "Some number plus 52 is at least 100."
What number plus 52 is at least 100?
Now uncover the term.
52 2:: 100
Think: 3 times some number is at least 48.
3x 2::
3 times ___ equals 48.
x 2:: c Method 2: Solve the inequality by undoing the operations. Step 1: Make a table. First, list the
operations in
the inequality
according to
the order in
which they
are applied to
the variable
Operations in
the Inequality
To Solve
1. First x is
V
by 3.
/
2. Then, 52 is /
I
Step 2: Apply the steps
in the "to solve" column to
solve the inequality.
1. First
52 from both sides
of the inequality.
2. Then
both sides by 3.
Then, starting with
the last operation
in the inequality
write the opposite of
the step. Continue
writing the opposite
until every step is
accounted for.
3x + 52 2:: 100
3x + 52 2:: 100
x 2::
You must make at least
Chapter 11
sales.
493 Lesson 7
1a. How can you check your solution?
1b. Would the graph of the solution set be a ray or individual points? Explain your answer.
Solving Two-Step Inequalities Containing Fractions
Solve
:4 - 5 < -2. Then graph the solution set.
Complete the table and solution steps.
Solution
...£-5
<-2
-4
1. First _ _ _ _ _ _ _ __
to both sides of the inequality.
...£-5
<-2
-4
2. Then, __________
3(-4)
both sides by _ _ __
and reverse the inequality
mbo!.
x>
Graph the solution on a number line. Put an _____ circle on -12, since the inequality
sign is greater than, not greater than or equal to. Then, the ray goes to the ________
0(
I
-14
I.
-13
-12
-11
-10
-9
-8
Solve each inequality.
2a.
-13>~-3
2b. 40:S; -3x + 10
2d. How is solving inequalities different from solving equations?
Chapter 11
494
Lesson 7
EXAMPLE
Cathy has $100 saved to spend on clothes. She wants to purchase a
winter jacket for $40 and some sweaters that cost $20 each. How many
sweaters can Cathy buy?
A
Write an inequality that represents the situation.
Operations in
the Inequality
).
1. First x is
J
2. Then,
/
,
To Solve
/
First
from both sides of the
inequality.
Solution
20x + 40:::; 100
20x+ 40
:::; 100
20x
20
2. Then
60
20
x<
both sides by
Cathy can buy ____~_ _ _ _ _"_ _ _ _ _ _ __
sweaters.
Since it is / is nOt possible to buy a negative number of sweaters, a graph of the
solution set will /
not include values less than O. Cathy could not have
bought part of a sweater so the graph is a _ _ _ _ _ _ _ _ _ _ _ _ __
B
Graph the solution set.
0(
I)
I
-2
-1
o
2
3
4
TRY THIS!
3a. A CD costs $12 and a DVD costs $15. You have $60. You plan to buy 1 DVD
and some CDs. Write and solve an inequality to determine how many CDs
you can buy. What does the solution mean in this situation?
3b.
Would you use a ray or a set of points for this solution? What is the
solution set? Explain.
Chapter 11
495 Lesson 7
Solving Multi-Step Inequalities
Ms. Vega plans to spend no more than $120 on 4 gallons of paint. The paint at a home improvement store is on sale for $5 off each gallon. Solve the inequality 4(r - 5) S 120 for r, the regular price per gallon of the paint Ms. Vega can afford. Method 1: Solve the inequality by first applying the Distributive Property.
4(r- 5) ~ 120
• r-
•5
120
~
120
~
4r-
Distributive Property
Addition Property ofInequality
+20
+
~
4r
4r < -.140
Division Property ofInequality
4­
r<
Method 2: Solve the inequality by first applying the Division Property of Inequality.
4(r- 5) ~ 120
4(r -: 5) <
4
­
BtL
Division Property ofInequality
r-5~
+
+5
r
Addition Property ofInequality
~
Interpret the solution set.
What does the solution set tell you? Does it make sense for the regular price
per gallon to be $0 or less than $O? Explain.
Graph the solution set of the inequality.
•I
-10 -5
Chapter 11
I•
0
5
10
15
20
496
25
30
35
40
Lesson 7
REFLECT
4a. Could Ms. Vega afford to buy paint that regularly costs $28 per gallon?
Explain how you know.
TRY THIS!
Solve each inequality.
4b. 3(x + 2) > 15
4c.
4n
+ 7n -
5
<6
4d.
lOs - 3 2:: 2s
+ 29
Solve each inequality. Round to the nearest hundredth, if necessary.
1.
lOx
+ 4 2:: -6 4. ~ + II < 15 -:)
2.
-3x - 21 > 16
3•
~+l>Al
2
-~
5.
1.5x - 2 ::::: 16
6.
0.2> -1.2x - 5.1
-----~---.---
Solve each inequality. Then graph the solution set.
>,
c
'"0E
7.
y
8. 42 < -9 + 30
-5x - 17::::: 38 (3
01
.r::
~
:0
:l
el.
i::
:l
e
'"
:r:
.E'
:E
~
"'1
I
-12 -11 -10 -9 -8 -7 -6 -5 -4
9.
I.
-3 -2 -1
•l i t o
-113 -112 -111 -110 -109 -108 -107
0
Dominique has $5.00. Bagels cost $0.60 each and a small container of
cream cheese costs $1.50.
s::
t
:l
~
a. How many bagels can Dominique buy if she also buys one small
container of cream cheese? Explain your answer.
©
b. Graph the solution set.
Chapter 11
I to
•I
-2
-1
497 0
2
345
Lesson 7
Yasmine and Alex each have $200 to spend
on clothes. Use the table for 10-11.
10. Yasmine decides to purchase a jacket and
some long-sleeve shirts. How many
long-sleeve shirts caB she buy?
Short-sleeve shirt
15
Long-sleeve shirt
20
Pair of jeans
30
Jacket
50
11. Alex wants to buy a jacket, 2 long-sleeve shirts, and some short-sleeve shirts.
Can she buy at least 8 short-sleeve shirts? Explain.
Solve each inequality.
12. 6(v - 4) < 12
13. -2(t + 5) 2: lO
14. 7x - 4x + 9 > 18
15. 3m + 7m - 0.8 ::5 9.2
16. 8g + 4 < 5g + 22
17. c + 8 > 5c - 28
Solve each inequality, and explain what the solution set means in the
context of the situation.
18. Renting a tent from a camping store costs $15 for the first day and $5 for each additional day.
Nell wants to spend no more than $30 to rent a tent. The inequality 15 + 5ed - 1) ::5 30
represents this situation, where d is the number of days Nell can rent the tent without
going over her budget.
19. Cora Plumbing charges a $25 fee plus $45 per hour. Atera Plumbing charges a $40 fee plus $40 per hour. The inequality 25 + 45h < 40 + 40h can be used to find h, the number of hours of work for which Cora Plumbing would be less expensive that Atera Plumbing. 20. Error Alert Vince solved the inequality 4(x + 5) > 20 as
shown. What error did Vince make? What is the correct
solution set of the inequality?
If(x + 5) > 20 Ifx + 5 > 20 If
If
x + 5 > 20
_ _-.;:..5
x
Chapter 11
498 -=2
> 15
Lesson 7
11-1 -
Additional Practice
Solve. Then graph each solution set on a number line.
1. 5x
8 < 17 _"~""~"""""~""~""""""""_
3. 3n + 7 - 7n <-5
5. 2(w + 36) + 10> 32 _""_~_~"_"~_~
u
6. -2 - 5;:;-9
----~-~-~~
Solve.
7. -7d+ 8> 29
10.2(-a-8)-15<9
8. 4(g - 3) + 1 ;:; 5
c
11.9+-;:;17
6
9. 12 - 3b < 9
1
12. -p-8-p:?.4
3
--~--"-----
13. Fifty students in the seventh grade are trying to raise at least $2,000 for sports
supplies. They have already raised $750. How much should each student raise, on
average, in order to meet the goal?
Chapter 11
499
Practice and Problem Solving
Write the correct answer.
1. Grace earns $7 for each car she
washes. She always saves $25 of her
weekly earnings. This week, she
wants to have at least $65 in
spending money. At least how many
cars must she wash?
2. Monty has saved $400 to spend on a
video game player and games. The
player he wants costs $275. The
games each cost $39. At most, how
many games can he buy along with
the player?
3. A video game club charges $8 per
month as a membership fee, plus
$2.75 for each game rental. Eugenie
plans to join and rent no more than
5 games a month. What amount
should she budget each month for
video games?
4. Cooper Middle School has a goal of
collecting more than 1,000 cans of
food in a food drive. So far, 375 cans
have been collected. During the last
13 days of the drive, at least how
many cans must be collected each
day in order to meet the goal?
Choose the letter for the best answer.
5. In January 2005, it cost $0.37 to mail
a letter weighing up to 1 ounce. Each
additional ounce or part of an ounce
cost $0.23. At most, what is the
weight of a letter with $1.06 in
postage?
A
w < 4 oz
B w < 3 oz
C
w ~ 4 oz F n~ 7
G n< 7
D w ~ 3 oz 7. The 12 members of the Middle
School filmmaking club need to raise
at least $1,400 to make a short film.
They already have raised $650. How
much more should each member
raise on average?
A
x 2 $62.50
B x~ $62.50
Chapter 11 C
6. Martin is planning a hedge along the
back of his yard. The total length can
be no more than 23 feet, and he will
put a 4-foot-wide gate in the hedge.
Each plant needs 2.5 feet of space to
grow properly. How many plants
should he buy?
H
n~9
J
n<9
8. The rule of thumb in filmmaking
is that you must shoot at least
3 minutes of film for every minute in
a movie's "final cut." A 30-minute roll
of film costs $250. How much will film
cost to make a 90-minute movie?
x < $62.50 D x> $62.50 500
C~
F C2 $22,500
H
G C ~ $7,500
J C2 $2,250
$67,500
Practice and Problem Solving
connections
(~~~'MON
Aiden is playing FishtoWll, a game where players
set up a virtual fish tank, raise the fish, and then sell the fish to
earn virtual dollars. The goal is to move up to the next level by
selling fish and earning at least $150. Which fish should Aiden
raise, and how many fish should he sell in order to move up to
the next level?
FishtoWll players start with $100. Players must use the
money to buy fish and decorate their tank. The table
shows the prices of different types of fish.
A
IS ,
,
Aiden plans to buy some tetras. Write an expression
that represents the amount of money that remains
after buying t tetras.
CORE
........... "
CC7.EE.4
CCl.EE.4a
CC.7.EEAb
Fishtown Price List
Type of Fish
Price
Angelfish
$8
Guppy
$3
Tetra
$2
Catfish
$5
Aiden wants to have at least $80 left after buying the tetras. Write and solve
an inequality to determine the number of tetras Aiden can buy. Show your
work.
C What are the different numbers of tetras Aiden could buy?
D
Aiden buys the greatest possible number of tetras. Next, he wants to buy
guppies. Write an expression that represents the amount of money that
remains after buying g guppies.
Chapter 11 501
Problem Solving Connections
.1 Aiden wants to be sure that he has at least $25 left after buying the guppies.
Write and solve an inequality to determine the number of guppies Aiden can
buy. Show your work.
Aiden buys the greatest possible number of guppies. How many guppies
does he buy? What is the total amount he has spent so far on fish?
D
Decorate the Tank
Aiden wants to use his remaining money to decorate his
tank. The table shows the prices of decorations.
/4... How much money does Aiden have for decorations?
Decoration
Price
Plant
$2.50
Castle
$4.25
Treasure chest
$3.25
.Pirate ship
,
$2.75
Aiden decides to buy a pirate ship and a treasure chest. He plans to spend
the rest of his money on plants. Write an expression that represents the total
cost of decorations if Aiden buys p plants.
Write and solve an equation to determine the number of plants Aiden can buy. Show
your work.
502 /
Fish Population Growth Rates
Once the tank contains fish, new ones are added every
hour. This increases the fish population. Different
types of fish have different population growth rates,
as shown in the table.
A
B
Write an expression that represents the total number of tetras in Aiden's tank after h hours. Type of Fish
Rate
Angelfish
1 new fish per hour
Guppy
7 new fish per hour
Tetra
i
Catfish
l
9 new fish per hour
3 new fish per hour
How long will it take until Aiden has more than 80 tetras in his tank?
Explain. (Assume new fish are added to the tank only at the end of each
hour.)
C Write an expression that represents the total number of guppies in Aiden's
tank after h hours.
0
In order to sell fish, the tank must have the same number of two different
types of fish. Write and solve an equation to determine how many hours
Aiden should wait until he has the same number of tetras and guppies. Show
your work.
E
How many tetras and how many guppies will Aiden have in his tank at this
time?
>. c
'":l.
E
0
\.J
01
c:
~
:D
::J
c..
1::
::J
0
:::
'"
:J:
.S
1i:
~
c:
.s
.<::
01
::J
Q
:c
@
Chapter 11 503
Problem Solving Connections
DAnswer the Question
It's time for Aiden to sell some fish! He needs to raise at least $150 from the sale of
the fish in order to move up to the next level of the game .
.'/f;.. . The table shows the sale prices for the fish. Note
that every time a player sells a type of fish, there is a
transaction fee. This fee depends only on the type of
fish sold, not the quantity sold. Write an expression
that shows the amount of money Aiden earns by
selling x tetras.
·./.'... ···!...·....ii5;QdJJ~~"',,/.g;i
•.............:.f··
•..".
.• ·..,;:c.
~"'c :~···
'TYpe of
Fish
Sale
Price
Transaction
Fee
Angelfish
$7.25
$9
Guppy
$2.50
$15
Tetra
$1.75
$18
Catfish
$3.75
$8
}
.' II Write and solve an inequality to determine whether Aiden can make at least $150 by selling only tetras. .....•.
.-:' Write and solve an inequality to determine whether Aiden can make at least
$150 by selling only guppies.
[) Suppose Aiden sells 42 guppies. Is it possible for him to sell some of his
tetras in addition to these guppies in order to make at least $1501 If so, how
many tetras should he sell? Explain.
"
504 ..
Name
Pe
('~~~MON
.
1.
*
2.
'
The base of an isosceles triangle is 22 inches. If the perimeter of
the triangle is 60 inches, what are the lengths of the sides of
the triangle? Show your work. CORE
........... COEE.1
COEE.4
CC.7.EE.4<i
CC7.EE.4b
Yolanda saves money to buy a $200 cell phone. She has $70, and saves $25
each week.
a. Write and solve an inequality to find out how many full weeks Yolanda
must save to have at least enough money to buy the cell phone.
b. Yolanda buys the phone when she has enough money saved.
Will she also have enough to buy a $35 protective case? Explain.
Samir used to make $0.40 more per hour than Jane. However,
Jane got a 15% raise, and now she makes $9.20 per hour.
a. Samir wants to know how much money Jane got for the raise.
He vvrites the equation J + 0.15J = 9.20. Solve this equation.
What does the solution mean?
Chapter 11 505
Performance Task
b. How much was Jane's raise? How do you know?
c. How much more does Jane now make than Samir? Explain .
.....
. . 4.
..
...
Yummy Snacks sells 36-ounce bags of mixed nuts. The bags have
peanuts, cashews, walnuts, and almonds. At most, of the nuts
are peanuts. The rest of the bag has an equal amount of each other
type of nut.
i
a. Write an inequality to represent the ounces of peanuts in a bag.
b. Write and solve an equation to represent the ounces w of walnuts
in a bag if there are p ounces of peanuts. Use your equation and
the limit on the number of peanuts in the bag to write an inequality
that shows how many ounces of walnuts there are in a bag. Explain
how you found your inequality.
506 PerTOrman!l)e Task
Date ____
Class _ _ _ __
Name
1. A mover notes the weights of a table and
4 chairs and records t + 4c ~ 100 on his
invoice. What is he communicating?
5. Mrs. Hughes' class has 22 students. Her
principal tells her that her class will
increase to 30 students. Which equation
can be used to find the percent increase?
A. The table and 4 chairs each weigh
more than 100 pounds.
A. 22
B. 22 =
B. The table and 4 chairs weigh at most
100 pounds.
D. 30
2. Hannah has $175 to spend. She buys $120
worth of non-taxable items. Some other
items are taxable at 6%. Which inequality
shows how much she can spend on
taxable items before tax is applied?
F. 1.25s ~ 10.00
G. 1.25s
J. ~l~--;;;
::; 10.00
._;:,
H. x S $51.89
7. The price of mailing a small package is
$0.32 for the first ounce and $0.21 for each
additional ounce. Sandra paid $1.16 to
mail her package. How much did it weigh?
J. x s: $165.09
I Skateboard
Price ($)
Go Green
45
Speedster
47
Up and Down
43
41
With the Flow
I
A. Go Green
c.
B. Speedster
D. With the Flow
~
Chapter 11
$6.69
Up and Down
~
C. 6 ounces
B. 5 ounces
D. 7 ounces
F. 2.7 -
1.8x = 2
G. 1.8x
2 = 2.7
H. 2x - 1.8 = 2.7
J. 2.7 - 2x = 1.8
9. Solve 5h + 15 - 3h = 32.
A. h = 16
H. xS $13.38
J. x
A. 4 ounces
8. A bench is being centered on a wall.
The wall is 2.7 m long and the bench is
1.8 m wide. \Vhich equation can be used
to determine how much of the wall should
be on each side of the bench?
4. Ken has $18 to spend on two models of the
solar system and supplies to paint them.
The two models cost the same amount.
His paint supplies cost $4.62. Wl1ich
expression indicates how much he can
spend on each model?
G. x
s: 10.00
H. 1.S2-=;:, ~ 10.00
3. Brad bought a skateboard for $2 less than
half its original price. If he paid $21.50,
which skateboard did he buy?
F. x S $6.69
22x = x
6. \Vhich inequality can be used to find how
many $1.25 snack packs can be purchased
for SlO.OO?
D. The table and 4 chairs weigh at least
100 pounds.
G. x S $45.09
30x
C. 22 + 22x = 30
C. The table and 4 chairs weigh around
100 pounds, give or take a little.
f. x S $3.30
+ X= 30
B. h
$13.38
501 = 23~
C. h = 8~
D. h
= 2fl
Assessment Readiness
10. Solve 2(a - 5) - 5 = 3. 15. A baseball stadium has 37,101 seats in the
three areas listed in the table.
H. a =-9
F. a=9
J. a=-12
G. a = 12
11. Juan needs to take a taxi to get to the
movies. The taxi charges $3.50 for the first
mile, and then $2.75 for each mile after
that. If the total charge is $18.63, then how
far was Juan's taxi ride to the movie?
A. 6.5 miles
C. 6.8miles
B. 5.3 miles
D. 5.5 miles
12. Solve 6(s-8)
~
F. s ~ -5
G. s <
- -.§.
3
Suppose all the box level and lower deck
seats during a game are filled. Write and
solve an inequality to determine how
many people could be sitting in the
upper deck.
-18
H. s ~ 5
J.
s~-ll
16. Katia has one more than five times the
number ofwristbands that Shelly has.
Rae has three more than twice the number
that Shelly has. What expression would
show how many more wristbands Katia
has than Rae? Show your work.
13. Larry has $389.00. A DVD player costs
$97.00, and he can purchase used movies
for $11.55 each. What is the greatest
number of movies Larry can buy if he also
buys a DVD player?
A. at most 26
C. at most 25
B. at most 34
D. atmost43
CONSTRUCTED RESPONSE
14. Henry is putting a new baseboard
around his room. He used the formula
P 2(£ + w) to find the perimeter. The
perimeter is 72~ feet. He remembers that
the width was 16~ feet. Show two different
ways to find the length of the other wall.
=
Chapter 11 17. Lacey has $20 to spend on school
supplies. Notebooks cost $2.50, pens
cost $0.50 and pencils cost $0.12. Lacey
needs 7 notebooks for her classes and
also wants to get 4 pens. How many
pencils cans she buy? Explain.
508
Assessment Readiness