College Algebra Chapter 1 Equations and Inequalities Section 1.7 Linear, Compound, and Absolute Value Inequalities Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Solve Linear Inequalities in One Variable Solve Compound Linear Inequalities Solve Absolute Value Inequalities Solve Applications of Inequalities

Concept 1 Solve Linear Inequalities in One Variable

Solve Linear Inequalities in One Variable Properties of Inequality Let a, b, and c represent real numbers. If x < a, then a > x If a < b, then a + c < b + c If a < b, then a – c < b – c If c is positive and a < b, then If c is negative and a < b, then These statements are also true expressed with the symbols ≤, >, and ≥.

Example 1 Solve the inequality. Graph the solution set, and write the solution in set-builder notation and interval notation. 3(x + 4) > 5x - 8

Example 2 Solve the inequality. Graph the solution set, and write the solution in set-builder notation and interval notation. 16 ≤ 8(x – 2) – 3x + 5

Skill Practice 1 Solve the inequality. Graph the solution set and write the solution set in set-builder notation and in interval notation. -5t – 6 ≥ 24

Example 3 Solve the inequality. Graph the solution set, and write the solution in set-builder notation and interval notation. 0.3(x - 4) < 0.5(x - 6)

Example 4 Solve the inequality. Graph the solution set, and write the solution in set-builder notation and interval notation.

Skill Practice 2 Solve the inequality. Graph the solution set and write the solution set in set-builder notation and in interval notation.

Example 5 Solve the inequality. Graph the solution set, and write the solution in set-builder notation and interval notation. 3(x – 3) + 17 ≤ 2(x + 4) + x

Example 6 Solve the inequality. Graph the solution set, and write the solution in set-builder notation and interval notation.

Concept 2 Solve Compound Linear Inequalities

Solve Compound Linear Inequalities “and” means intersection “or” means union Three-part compound inequality (x is between two values): the goal is to isolate x in the middle region Note: A three-part inequality is used to imply that x is between two values. -3 < x < 5  Yes -3 < x > 5  No

Example 7 Solve the inequality. Graph the solution set, and write the solution in set-builder notation and interval notation. 2x > 10 or 2x – 3 < 5

Skill Practice 3 Solve.

Example 8 Solve the compound inequality. Graph the solution set, and write the solution in set-builder notation and interval notation. 6(x – 2) ≤ 10x and 5x < 10

Skill Practice 4 Solve

Example 9 Solve the compound inequality. Graph the solution set, and write the solution in set-builder notation and interval notation.

Example 10 Solve the compound inequality. Graph the solution set, and write the solution in set-builder notation and interval notation.

Example 11 -1 ≤ 2x – 3 < 5

Example 12

Example 13 -3 < 1 – 4y < -1

Skill Practice 5 Solve. 16 ≤ -3y – 4 < 2

Concept 3 Solve Absolute Value Inequalities

Solve Absolute Value Inequalities If k is real number, k>0, then |u| < k is equivalent to –k < u < k (also holds for ≤) Example: |x| < 3 If k is a real number, k >0, then |u| > k is equivalent to u < -k or u > k (also holds for ≥) Example: |x| < 3

Example 14 Solve |2x + 3| < 5

Example 15 Solve |2x + 5| + 2 ≤ 11

Skill Practice 6 Solve the inequality and write the solution in interval notation. 3|5 – x| + 2 ≤ 14

Example 16 Solve 5|2x – 3| - 1 > 9

Example 17 Solve

Skill Practice 7 Solve the inequality and write the solution in interval notation. -18 > -3|2y – 4|

Example 18 Solve |2x + 1| ≥ 0

Example 19 Solve |6 – 5x| < -4

Skill Practice 8 Solve |x – 3| < -2 |x – 3| > -2 |x + 1| ≤ 0

Concept 4 Solve Applications of Inequalities

Example 20 The pH of the water in a public swimming pool should be maintained at a safe swimming value of 7.4. Slight variations in the tested value of the pH levels are acceptable but should differ from the ideal pH level by no more than 0.2. If x represents the exact pH value tested, write an absolute value inequality that represents a safe interval for x. Solve the inequality and interpret the answer.

Example 21 One cell phone service charges a flat rate of $35 a month plus1¢ per text. Another company offers a flat monthly fee of $50 with unlimited texting. How many texts would you need to send for the first company to charge you more per month than the second company?

Skill Practice 9 For a recent year, the monthly snowfall (in inches) for Chicago, illinois, for November, december, January, and february was 2,8.4, 11.2, and 7.9, respectively. How much snow would be necessary in march for Chicago to exceed its monthly average snowfall of 7.28 in. for these five months?

Skill Practice 10 A board is to be cut a length of 24 in. the measurement error is no more than 0.02 in. Let x represent the actual length of the board. Write an absolute value inequality that represents an interval in which to estimate x. Solve the inequality from part (a) and interpret the meaning.