Page 1 of 4
Solving Inequalities
BEFORE
Now
WHY?
You wrote and solved
equations.
You’ll write and solve
inequalities.
So you can determine the better
deal, as in Ex. 38.
In the Real World
Word Watch
inequality, p. 346
solution of an inequality,
p. 346
graph of an inequality,
p. 346
equivalent inequalities,
p. 346
Bumper Cars You must be at least
42 inches tall to ride the bumper
cars. This can be represented
by the inequality h ≥ 42. An
inequality is a mathematical
sentence formed by placing an
inequality symbol between two
expressions. The solution of an
inequality is the set of numbers
that you can substitute for the
variable to make the inequality true.
The graph of an inequality in one variable is the set of points on a
number line that represents the solution of the inequality.
EXAMPLE
with
1
Graphing Inequalities
a. x < 2
All numbers less than 2
b. x ≤ 1
All numbers less than
or equal to 1
c. x > 0
All numbers greater than 0
d. x ≥ 2
All numbers greater than
or equal to 2
Reading
The inequality symbol ≤ is
read “is less than or equal
to.” The inequality ≥ symbol
is read “is greater than or
equal to.”
3 2 1 0 1 2 3
3 2 1 0 1 2 3
3 2 1 0 1 2 3
3 2 1 0 1 2 3
Equivalent inequalities are inequalities that have the same solution.
You can produce an equivalent inequality in the following ways.
• Add or subtract the same number on each side.
• Multiply or divide each side by the same positive number.
1<3
Multiply each side by 2.
2<6
• Multiply or divide each side by the same negative number and reverse
the direction of the inequality.
1<3
346
Chapter 7
Equations, Inequalities, and Functions
Multiply each side by 2.
2 > 6
Page 2 of 4
EXAMPLE
2
Solving an Inequality
Solve d 2 ≤ 1. Then graph the solution.
d2≤1
Write original inequality.
d22≤12
d≤3
Add 2 to each side.
Simplify.
To graph d ≤ 3, use a closed dot and draw the arrow pointing to
the left.
2
1
0
1
2
3
4
5
✓Check
To check the solution d ≤ 3, choose any number less than
or equal to 3 to substitute for d. The check below uses d 0.
d2≤1
0 2?
≤ 1
2 ≤ 1 ✓
Your turn now
Write original inequality.
Substitute 0 for d.
Solution checks.
Solve the inequality. Then graph the solution.
1. x 3 ≥ 1
tch Out!
Wa
EXAMPLE
3
2. 6 < t 5
3. w 9 < 4
Solving an Inequality
Solve 4w < 20. Then graph the solution.
Don’t forget to
reverse the inequality when
you multiply or divide each
side of an inequality by a
negative number.
4w < 20
Write original inequality.
4w 20
> 4
4
Divide each side by 4. Reverse inequality.
w > 5
Simplify.
To graph w > 5, use an open dot and draw the arrow pointing to
the right.
10
Your turn now
4. 3n ≥ 24
5
0
5
10
Solve the inequality. Then graph the solution.
5. 6s < 42
Lesson 7.6
1
6. x ≤ 5
2
Solving Inequalities
347
Page 3 of 4
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 711
Getting Ready to Practice
1. Vocabulary How is the graph of the solution of an inequality, such
as x 1 > 2, different than the graph of a solution of an equation, such
as x 1 2?
Graph the inequality.
2. a ≥ 12
3. x ≤ 7
4. w < 5
5. d > 13
Solve the inequality. Then graph the solution.
6. r 17 ≤ 21
7. b 17 ≥ 8
x
9. < 4
3
8. 5y ≥ 35
10. Writing When do you use a closed dot in graphing an inequality?
When do you use an open dot?
Practice and Problem Solving
with
Example
1
2
3
Homework
Exercises
15–18
22–33
22–33
Tell whether the value of m is a solution to the inequality m ≤ 5.
11. 1
12. 5
13. 10
14. 4
Write an inequality represented by the graph.
15.
Online Resources
1
16.
0
1
2
3
4
10 5
0
5
10 15
3 2 1
0
1
2
6
8
10
CLASSZONE.COM
• More Examples
• eTutorial Plus
17.
18.
0
2
4
Write an inequality to represent the situation. Then graph the inequality.
19. To vote in the United States, a person must be 18 years old or older.
20. Children under 12 will be admitted to the museum at no charge.
21. In football, you need at least 10 yards for a first down.
Solve the inequality. Then graph the solution.
348
Chapter 7
22. p 7 < 13
23. t 14 ≥ 28
24. w 21 ≤ 32
25. 19 e > 41
26. 56 y ≥ 113
x
1
27. ≥ 9
3
b
28. > 5
4
c
29. < 17
8
30. 3g > 36
31. 7s ≤ 42
32. 27n < 108
33. z 8.6 > 16.4
Equations, Inequalities, and Functions
Page 4 of 4
In Exercises 34–37, solve the inequality and graph the solution. Then
check your answer.
Solving a Two-Step Inequality
EXAMPLE
2x 1 ≤ 4
Original inequality
2x 1 1 ≤ 4 1
2x ≤ 3
Simplify.
2x 3
≤ 2
2
3
x ≤ 2
34. 3 6x ≤ 21
Subtract 1 from each side.
Divide each side by 2.
Simplify.
35. 2b 13 > 9
36. 2y 5 > 15 37. 7 4x ≤ 9
38. Subway Pass A one-way subway trip costs $.75. A monthly subway
pass costs $27. Write and solve an inequality to find the least number of
one-way rides you must take for the subway pass to be a better deal
than paying by the ride.
39. Writing Write a real-world problem that can be modeled by 3 x ≤ 10.
Challenge Describe all of the numbers that satisfy both inequalities.
40. 4x > 2 and 2.5x 4 ≤ 9
41. 2y 7 < 21 and 25y 2 > 23
Mixed Review
42. Find the greatest common factor of 15, 20, and 30. (Lesson 4.2)
43. Solve the equation 8 9x 73. (Lesson 7.5)
Basic Skills Find the sum or difference.
44. 9 (15)
45. 22 (13) 46. 7 (3)
47. 8 (9)
Test-Taking Practice
INTERNET
State Test Practice
CLASSZONE.COM
48. Multiple Choice Which is the solution of the inequality represented by
“3 more than twice a number is greater than 20”?
A. n < 11.5
B. n > 11.5
C. n > 13
D. n < 7
49. Short Response A boat lift can lift up to 3600 pounds. An empty boat
on the lift weighs 2300 pounds. Write and solve an inequality to find
the number of additional pounds allowed in the boat without
exceeding the lift’s capacity.
Lesson 7.6
Solving Inequalities
349