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

Vocabulary compound inequality intersection union

Concept

Example 1 Solve an “And” Compound Inequality Solve 10  3y – 2 < 19. Graph the solution set on a number line. Method 1Solve separately. Write the compound inequality using the word and. Then solve each inequality. 10  3y – 2and3y – 2 <  3y3y < 21 4  y y < 7 4  y < 7

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

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

Example 1 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 .

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

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

Concept

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

Example 2 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  –4 or x + 3<2 x<–1 x4x4

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

Example 3 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: 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.

Example 3 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.

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

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

Example 3b 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 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}

Concept

Example 4 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  4or2x – 2  –4 2x  62x  –2 x  3x  –1 Answer:

Example 4 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  4or2x – 2  –4 2x  62x  –2 x  3x  –1 Answer: The solution set is  x | x  –1 or x  3 .

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

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

Example 5 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: The starting salary can differ from the average by as much as$2450. |38,500 – x|  2450

Example 5 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

Example 5 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 Write and Solve an Absolute Value Inequality Answer:

Example 5 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.