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 and3y – 2 < 19

12 3y3y < 21

4 y y < 7

4 y < 7

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

Solve an “And” Compound Inequality

Graph the solution set for each inequality and find their intersection.

y 4

y < 7

4 y < 7

Answer: The solution set is y | 4 y < 7.

What is the solution to 11 2x + 5 < 17?

A.

B.

C.

D.

Solve an “Or” Compound Inequality

Solve x + 3 < 2 or –x –4. Graph the solution set on a number line.

Answer: The solution set is x | x < –1 or x 4.

x < –1

x 4

x < –1 or x 4

Solve each inequality separately.

–x –4orx + 3 < 2

x < –1 x 4

What is the solution to x + 5 < 1 or –2x –6?Graph the solution set on a number line.

A.

B.

C.

D.

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.

Answer: The solution set is d | –2 < d < 2.

All of the numbers between –2 and 2 are less than 2 units from 0.

Notice that the graph of 2 > |d| is the same as the graph of d > –2 and d < 2.

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.

Answer: The solution set is d | d < –3 or d > 3.

All of the numbers not between –3 and 3 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.

A. What is the solution to |x| > 5?

A.

B.

C.

D.

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}

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.

What is the solution to |3x – 3| > 9? Graph the solution set on a number line.

A.

B.

C.

D.

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.

Answer: |38,500 – x| 2450

The starting salary can differ from the average by as much as $2450.

|38,500 – x| 2450

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 –36,05040,950 x 36,050

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.

A. |4.5 – w| 7

B. |w – 4.5| 7

C. |w – 7| 4.5

D. |7 – w| 4.5

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. {w | w ≤ 11.5}

B. {w | w ≥ 2.5}

C. {w | 2.5 ≤ w ≤ 11.5}

D. {w | 4.5 ≤ w ≤ 7}

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?