Solving Linear Inequalities Section 1.6 Solving Linear Inequalities

Objectives Solve linear inequalities. Solve compound inequalities. Solve applied problems using inequalities.

Inequalities An inequality is a sentence with <, >, , or  as its verb. Example: 3x  5 < 6 – 2x To solve an inequality is to find all values of the variable that make the inequality true. Each of these numbers is a solution of the inequality, and the set of all such solutions is its solution set. Inequalities that have the same solution set are called equivalent inequalities.

Principles for Solving Inequalities For any real numbers a, b, and c: The Addition Principle for Inequalities: If a < b is true, then a + c < b + c is true. The Multiplication Principle for Inequalities: If a < b and c > 0 are true, then ac < bc is true. If a < b and c < 0, then ac > bc is true. Similar statements hold for a  b. When both sides of an inequality are multiplied or divided by a negative number, we must reverse the inequality sign.

Example Solve each of the following. Then graph the solution set. a. 3x – 5 < 6 – 2x b. 13 – 7x ≥ 10x – 4

Compound Inequalities When two inequalities are joined by the word and or the word or, a compound inequality is formed. Conjunction contains the word and. Example: 3 < 2x + 5 and 2x + 5  7 The sentence –3 < 2x + 5 ≤ 7 is an abbreviation. Disjunction contains the word or. Example: 2x – 5  –7 or 2x – 5 > 1

Example Solve – 3 < 2x + 5 ≤ 7. Then graph the solution set.

Example Solve: 2x – 5 ≤ –7 or 2x – 5 > 1. Then graph the solution set.

Application - Example For her interior decorating job, Natalia can be paid in one of two ways: Plan A: $250 plus $10 per hour Plan B: $20 per hour Suppose that a job takes n hours. For what values of n is plan B better for Natalia?

Example (continued) Solution: 1. Familiarize. Read the problem. For a 20 hour job, n = 20: Plan A: $250 + $10 • 20 = $450 Plan B: $20 • 20 = $400 Plan A is better for a 20 hour job. For a 30 hour job, n = 30: Plan A: $250 + $10 • 30 = $550 Plan B: $20 • 30 = $600 Plan B is better for a 30 hour job.

Example (continued) 2. Translate. Translate to an inequality. Income from plan B is greater than Income from plan A 20n > 250 + 10n 3. Carry out. Solve the inequality.

Application continued 4. Check. For n = 25, the income from plan A is $250 + $10 • 25, or $500, and the income from plan B is $20 • 25, or $500, the same under either plan. We have seen that plan B pays more for a 30-hr job. Since 30 > 25, this provides a partial check. We cannot check all values of n. 5. State. For values of n greater than 25 hr, plan B is better for Natalia.