1. Solving Compound and Absolute Value Inequalities
LESSON 1–6
Solving Compound and Absolute Value Inequalities

2. Five-Minute Check (over Lesson 1–5) TEKS Then/Now New Vocabulary
Key Concept: “And” Compound Inequalities
Example 1: Solve an “And” Compound Inequality
Key Concept: “Or” Compound Inequalities
Example 2: Solve an “Or” Compound Inequality
Example 3: Solve Absolute Value Inequalities
Key Concept: Absolute Value Inequalities
Example 4: Solve a Multi-Step Absolute Value Inequality
Example 5: Write and Solve an Absolute Value Inequality
Lesson Menu

3. Solve the inequality 3x + 7 > 22
Solve the inequality 3x + 7 > 22. Graph the solution set on a number line.
A. {x | x > 5}
B. {x | x < 5}
C. {x | x > 6}
D. {x | x < 6}
5-Minute Check 1

4. Solve the inequality 3(3w + 1) ≥ 4. 8
Solve the inequality 3(3w + 1) ≥ 4.8. Graph the solution set on a number line.
A. {w | w ≤ 0.2}
B. {w | w ≥ 0.2}
C. {w | w ≥ 0.6}
D. {w | w ≤ 0.6}
5-Minute Check 2

5. Solve the inequality 7 + 3y > 4(y + 2)
Solve the inequality 7 + 3y > 4(y + 2). Graph the solution set on a number line.
A. {y | y > 1}
B. {y | y < 1}
C. {y | y > –1}
D. {y | y < –1}
5-Minute Check 3

6. Solve the inequality . Graph the solution set on a number line.
A. {w | w ≤ –9}
B. {w | w ≥ –9}
C. {w | w ≤ –3}
D. {w | w ≥ –3}
5-Minute Check 4

7. A company wants to make at least $255,000 profit this year
A company wants to make at least $255,000 profit this year. By September, the company made $127,500 in profit. The company plans to earn, on average, $15,000 each week in profit. Will the company reach its goal by the end of the year?
A. yes
B. no
5-Minute Check 5

8. A2.6(F) Solve absolute value linear inequalities.
Targeted TEKS
A2.6(F) Solve absolute value linear inequalities.
Mathematical Processes
A2.1(E), A2.1(F)
TEKS

9. You solved one-step and multi-step inequalities.
Solve compound inequalities.
Solve absolute value inequalities.
Then/Now

10. compound inequality
intersection
union
Vocabulary

11. Concept

12. Solve 10  3y – 2 < 19. Graph the solution set on a number line.
Solve an “And” Compound Inequality
Solve 10  3y – 2 < 19. Graph the solution set on a number line.
Method 1 Solve separately.
Write the compound inequality using the word and. Then solve each inequality.
10  3y – 2 and 3y – 2 < 19
12  3y 3y < 21
4  y y < 7
4  y < 7
Example 1

13. Method 2 Solve both together.
Solve an “And” Compound Inequality
Method 2 Solve both together.
Solve both parts at the same time by adding 2 to each part. Then divide each part by 3.
10  3y – 2 < 19
12  3y < 21
4  y < 7
Example 1

14. Answer: The solution set is y | 4  y < 7.
Solve an “And” Compound Inequality
Graph the solution set for each inequality and find their intersection.
4  y < 7
y < 7
y  4
Answer: The solution set is y | 4  y < 7.
Example 1

15. What is the solution to 11  2x + 5 < 17?B. C. D.
Example 1

16. Concept

17. Solve each inequality separately. –x  –4 or x + 3 < 2 x < –1
Solve an “Or” Compound Inequality
Solve x + 3 < 2 or –x  –4. Graph the solution set on a number line.
Solve each inequality separately.
–x  –4 or
x + 3 < 2
x < –1
x  4
x < –1 or x  4
x < –1
x  4
Answer: The solution set is x | x < –1 or x  4.
Example 2

18. What is the solution to x + 5 < 1 or –2x  –6
What is the solution to x + 5 < 1 or –2x  –6? Graph the solution set on a number line. A. B. C. D.
Example 2

19. A. Solve 2 > |d|. Graph the solution set on a number line.
Solve Absolute Value Inequalities
A. Solve 2 > |d|. Graph the solution set on a number line.
2 > |d| means that the distance between d and 0 on a number line is less than 2 units. To make 2 > |d| true, you must substitute numbers for d that are fewer than 2 units from 0.
Notice that the graph of 2 > |d| is the same as the graph of d > –2 and d < 2.
All of the numbers between –2 and 2 are less than 2 units from 0.
Answer: The solution set is d | –2 < d < 2.
Example 3

20. B. Solve 3 < |d|. Graph the solution set on a number line.
Solve Absolute Value Inequalities
B. Solve 3 < |d|. Graph the solution set on a number line.
3 < |d| means that the distance between d and 0 on a number line is greater than 3 units. To make 3 < |d| true, you must substitute values for d that are greater than 3 units from 0.
Notice that the graph of 3 < |d| is the same as the graph of d < –3 or d > 3.
All of the numbers not between –3 and 3 are greater than 3 units from 0.
Answer: The solution set is d | d < –3 or d > 3.
Example 3

21. A. What is the solution to |x| > 5?B. C. D.
Example 3a

22. B. What is the solution to |x| < 5?
A. {x | x > 5 or x < –5}
B. {x | –5 < x < 5}
C. {x | x < 5}
D. {x | x > –5}
Example 3b

23. Concept

24. Solve |2x – 2|  4. Graph the solution set on a number line.
Solve a Multi-Step Absolute Value Inequality
Solve |2x – 2|  4. Graph the solution set on a number line.
|2x – 2|  4 is equivalent to 2x – 2  4 or 2x – 2  –4.
Solve each inequality.
2x – 2  4 or 2x – 2  –4
2x  6 2x  –2
x  3 x  –1
Answer: The solution set is x | x  –1 or x  3.
Example 4

25. What is the solution to |3x – 3| > 9
What is the solution to |3x – 3| > 9? Graph the solution set on a number line. A. B. C. D.
Example 4

26. The starting salary can differ from the average
Write and Solve an Absolute Value Inequality
A. JOB HUNTING To prepare for a job interview, Hinda researches the position’s requirements and pay. She discovers that the average starting salary for the position is $38,500, but her actual starting salary could differ from the average by as much as $2450. Write an absolute value inequality to describe this situation.
Let x = the actual starting salary.
The starting salary can differ from the average
by as much as
$2450.
|38,500 – x|  2450
Answer: |38,500 – x|  2450
Example 5

27. Write and Solve an Absolute Value Inequality
B. JOB HUNTING To prepare for a job interview, Hinda researches the position’s requirements and pay. She discovers that the average starting salary for the position is $38,500, but her actual starting salary could differ from the average by as much as $2450. Solve the inequality to find the range of Hinda’s starting salary. | 38,500 – x |  2450
Rewrite the absolute value inequality as a compound inequality. Then solve for x.
–2450  38,500 – x  2450
–2450 – 38,500  –x  2450 – 38,500
–40,950  –x  –36,050
40,950  x  36,050
Example 5

28. Write and Solve an Absolute Value Inequality
Answer: The solution set is x | 36,050  x  40,950. Hinda’s starting salary will fall within $36,050 and $40,950.
Example 5

29. A. HEALTH The average birth weight of a newborn baby is 7 pounds
A. HEALTH The average birth weight of a newborn baby is 7 pounds. However, this weight can vary by as much as 4.5 pounds. What is an absolute value inequality to describe this situation?
A. |4.5 – w|  7
B. |w – 4.5|  7
C. |w – 7|  4.5
D. |7 – w|  4.5
Example 5a

30. B. HEALTH The average birth weight of a newborn baby is 7 pounds
B. HEALTH The average birth weight of a newborn baby is 7 pounds. However, this weight can vary by as much as 4.5 pounds. What is the range of birth weights for newborn babies?
A. {w | w ≤ 11.5}
B. {w | w ≥ 2.5}
C. {w | 2.5 ≤ w ≤ 11.5}
D. {w | 4.5 ≤ w ≤ 7}
Example 5b

31. Solving Compound and Absolute Value Inequalities
LESSON 1–6
Solving Compound and Absolute Value Inequalities