LESSON
13.2
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Addition and
Subtraction Inequalities
Expressions,
equations, and
relationships—6.10.A
Model and solve one-variable,
one-step… inequalities that
represent problems. Also
6.9.B, 6.9.C, 6.10.B.
ESSENTIAL QUESTION
How can you solve an inequality involving addition
or subtraction?
6.10.A
EXPLORE ACTIVITY
Modeling One-Step Inequalities
You can use algebra tiles to model an inequality involving addition.
On a day in January in Watertown, NY, the temperature was 5 °F
at dawn. By noon it was at least 8 °F. By how many degrees did the
temperature increase?
A Let x represent the increase in temperature. Write an inequality.
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Alamy
Temperature
at dawn
+
Increase in
temperature
+
B The model shows 5 + x ≥ 8.
How many tiles must you remove
from each side to isolate x on
one side of the inequality?
≥
8
≥
8
+ + +
+ +
5
Circle these tiles.
+
+
≥
+ + + +
+ + + +
x
≥
8
C What values of x make this inequality true? Graph the solution of the
inequality on the number line.
x≥
-5 -4 -3 -2 -1
0 1 2 3 4 5
Reflect
1.
Analyze Relationships How is solving the inequality
5 + x ≥ 8 like solving the equation 5 + x = 8? How is it different?
Math Talk
Mathematical Processes
Could the temperature have
increased by 2 degrees
by noon? Could it have
increased by 5 degrees?
Explain.
Lesson 13.2
355
Using Properties of Inequalities
Addition and Subtraction Properties of Inequality
Math On the Spot
Addition Property of Inequality
my.hrw.com
Subtraction Property of Inequality
You can add the same number to You can subtract the same number
both sides of an inequality and the from both sides of an inequality
inequality will remain true.
and the inequality will remain true.
EXAMPLE 1
6.9.B, 6.10.B
Solve each inequality. Graph and check the solution.
A x + 5 < -12
STEP 1
Solve the inequality.
x + 5 < -12
5 ____
-5
____
x < -17
STEP 2
Use the Subtraction Property of Inequality.
Subtract 5 from both sides.
Graph the solution.
-20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10
STEP 3
Math Talk
Mathematical Processes
-13 < -12
B 8≤ y-3
STEP 1
Solve the inequality.
8 ≤y - 3
+
3
+
3
_
_
11 ≤ y
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Unit 4
The inequality is true.
Use the Addition Property of Inequality.
Add 3 to both sides.
You can rewrite 11 ≤ y as y ≥ 11.
STEP 2
Graph the solution.
STEP 3
Check the solution. Substitute a solution from the shaded
part of your number line into the original inequality.
5 6 7 8 9 10 11 12 13 14 15
?
8 ≤ 12 - 3
Substitute 12 for y in 8 ≤ y - 3
8≤ 9
The inequality is true.
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What would it tell you if the
inequality is false when you
check the solution?
Check the solution. Substitute a solution from the shaded
part of your number line into the original inequality.
?
-18 + 5 < -12 Substitute -18 for x into x + 5 < -12
YOUR TURN
Solve each inequality. Graph and check the solution.
2. y - 5 ≥ -7
Personal
Math Trainer
3. 21 > 12 + x
Online Assessment
and Intervention
my.hrw.com
-5 -4 -3 -2 -1
0 1 2
3 4 5
0 1 2 3 4 5 6 7
8 9 10
Interpreting Inequalities as Comparisons
You can write a real-world problem for a given inequality. Examine each
number and mathematical operation in the inequality.
EXAMPL 2
EXAMPLE
Math On the Spot
6.9.C
my.hrw.com
Write a real-world problem for the inequality 60 ≥ w + 5.
Then solve the inequality.
STEP 1
Examine each part of the inequality.
w is the unknown quantity.
5 is added to w.
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Robertson/Corbis
60 is greater than or equal to a number added to 5.
STEP 2
Write a comparison that the inequality could describe. June’s
dog will travel to a dog show in a pet carrier. The pet carrier
weighs 5 pounds. The total weight of the pet carrier and the
dog must be no more than 60 pounds. What inequality describes
the weight of June’s dog?
STEP 3
Solve the inequality.
60 ≥ w + 5
-5
____
55 ≥ w
-5
____
June’s dog currently weighs ≤ 55 pounds.
Reflect
4. If you were to graph the solution, would all points on the graph make
sense for the situation?
Lesson 13.2
357
YOUR TURN
Personal
Math Trainer
5. Write a real-world problem that can be modeled by x - 13 > 20. Solve
your problem and tell what values make sense for the situation.
Online Assessment
and Intervention
my.hrw.com
Guided Practice
1. Write the inequality shown on the model. Circle the tiles you would
remove from each side and give the solution. (Explore Activity)
Inequality:
+ + +
Solution:
+
≤
+ + + + +
Solve each inequality. Graph and check the solution. (Example 1)
2. x + 4 ≥ 9
0 1 2 3 4 5 6 7
3. 5 > z - 3
8 9 10
4. t + 5 > 12
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7
8 9 10
5. y - 4 < 2
0 1 2 3 4 5 6 7 8 9 10
?
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ESSENTIAL QUESTION CHECK-IN
7. Explain how to solve 7 + x ≥ 12. Tell what property of inequality you
would use.
358
Unit 4
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6. Write a real-world problem that can be represented by the inequality
y - 4 < 2. Solve the inequality and tell whether all values in the solution
make sense for the situation. (Example 2)
Name
Class
Date
13.2 Independent Practice
Personal
Math Trainer
6.9.B, 6.9.C, 6.10.A, 6.10.B
my.hrw.com
Online
Assessment and
Intervention
Solve each inequality. Graph and check the solution.
8. x - 35 > 15
0 10 20 30 40 50 60 70 80 90 100
9. 193 + y ≥ 201
0 1 2 3 4 5 6 7
8 9 10
10. y - 5 ≥ -15
- 12 - 11 - 10 - 9
-8
-7
-6
-5
-4
-3
-2
- 15 - 14 - 13 - 12 - 11 - 10 - 9
-8
-7
-6
-5
11. 15 ≥ z + 26
Write an inequality to solve each problem.
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12. The water level in the aquarium’s shark tank is always greater than 25 feet.
If the water level decreased by 6 feet during cleaning, what was the water
level before the cleaners took out any water?
13. Danny has at least $15 more than his big brother. Danny’s big brother has
$72. How much money does Danny have?
14. The vet says that Ray’s puppy will grow to be at most 28 inches tall. Ray’s
puppy is currently 1 foot tall. How much more will the puppy grow?
15. Pierre’s parents ordered some pizzas for a party. 4.5 pizzas were eaten at
the party. There were at least 5_12 whole pizzas left over. How many pizzas
did Pierre’s parents order?
16. To get a free meal at his favorite restaurant, Tom needs to spend $50 or
more at the restaurant. He has already spent $30.25. How much more
does Tom need to spent to get his free meal?
Lesson 13.2
359
17. Multistep The table shows Marco’s checking account
activity for the first week of June.
a. Marco wants his total deposits for the month of June
to exceed $1,500. Write and solve an inequality to find
how much more he needs to deposit to meet this goal.
Deposit – Paycheck
$520.45
Purchase – Grocery Store
$46.50
Purchase – Movie Theatre
$24.00
Purchase – Water bill
$22.82
b. Marco wants his total purchases for the month to be less than $450.
Write and solve an inequality to find how much more he can spend
and still meet this goal.
c. There are three weeks left in June. If Marco spends the same amount in
each of these weeks that he spent during the first week, will he meet his
goal of spending less than $450 for the entire month? Justify your answer.
FOCUS ON HIGHER ORDER THINKING
Work Area
19. Critical Thinking José solved the inequality 3 > x + 4 and got x < 1.
Then, to check his solution, he substituted -2 into the original inequality
to check his solution. Since his check worked, he believes that his answer
is correct. Describe another check José could perform that will show his
solution is not correct. Then explain how to solve the inequality.
20. Look for a Pattern Solve x + 1 > 10, x + 11 > 20, and x + 21 > 30.
Describe a pattern. Then use the pattern to predict the solution of
x + 9,991 > 10,000.
360
Unit 4
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18. Critique Reasoning Kim solved y - 8 ≤ 10 and got y ≤ 2. What might
Kim have done wrong?