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Unit 1 Expressions, Equations and Inequalities 1.7 Linear Inequalities and Absolute Value Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: • • • • • • Use interval notation. Find intersections and unions of intervals. Solve linear inequalities. Recognize inequalities with no solution or all real numbers as solutions. Solve compound inequalities. Solve absolute value inequalities. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Solving an Inequality Solving an inequality is the process of finding the set of numbers that make the inequality a true statement. These numbers are called the solutions of the inequality and we say that they satisfy the inequality. The set of all solutions is called the solution set of the inequality. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Interval Notation The open interval (a,b) represents the set of real numbers between, but not including, a and b. (a, b) x a x b The closed interval [a,b] represents the set of real numbers between, and including, a and b. [a, b] x a x b Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Interval Notation (continued) The infinite interval (a, ) represents the set of real numbers that are greater than a. (a, ) x x a The infinite interval ( , b] represents the set of real numbers that are less than or equal to b. (, b] x x b Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Parentheses and Brackets in Interval Notation Parentheses indicate endpoints that are not included in an interval. Square brackets indicate endpoints that are included in an interval. Parentheses are always used with or . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Using Interval Notation Express the interval in set-builder notation and graph: [1, 3.5] x 1 x 3.5 Express the interval in set-builder notation and graph: (, 1) x x 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Finding Intersections and Unions of Two Intervals 1. Graph each interval on a number line. 2. a. To find the intersection, take the portion of the number line that the two graphs have in common. b. To find the union, take the portion of the number line representing the total collection of numbers in the two graphs. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Finding Intersections and Unions of Intervals Use graphs to find the set: [1,3] (2,6) Graph of [1,3]: Graph of (2,6): Numbers in both [1,3] and (2,6): Thus, [1,3] (2,6) (2,3] Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Solving Linear Inequalities in One Variable A linear inequality in x can be written in one of the following forms a 0 : ax b 0 ax b 0 ax b 0 ax b 0 In general, when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol is reversed. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Solving a Linear Inequality Solve and graph the solution set on a number line: 2 3x 5 2 3x 5 3 x 3 3x 3 3 3 x 1 The solution set is x x 1 . The interval notation for this solution set is [ 1, ) . The number line graph is: Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Solving Linear Inequalities [Recognize inequalities with no solution or all real numbers as solutions] Solve the inequality: 3( x 1) 3 x 2 3( x 1) 3 x 2 3x 3 3x 2 32 The inequality is true for all values of x. The solution set is the set of all real numbers. In interval notation, the solution is (, ) In set-builder notation, the solution set is x x is a real number Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Solving a Compound Inequality Solve and graph the solution set on a number line: 1 2 x 3 11 Our goal is to isolate x in the middle. 1 2 x 3 11 In interval notation, 2 2 x 8 the solution is [-1,4). 1 x 4 In set-builder notation, the solution set is x 1 x 4 The number line graph looks like Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Solving an Absolute Value Inequality If u is an algebraic expression and c is a positive number, 1. The solutions of u c are the numbers that satisfy c u c 2. The solutions of u c are the numbers that satisfy u c uc or These rules are valid if is replaced by and is replaced by . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Solving an Absolute Value Inequality Solve and graph the solution set on a number line: 18 6 3x We begin by expressing the inequality with the absolute value expression on the left side: 6 3x 18 We rewrite the inequality without absolute value bars. 6 3x 18 means 6 3 x 18 or 6 3 x 18 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Solving an Absolute Value Inequality (continued) We solve these inequalities separately: 6 3 x 18 6 3 x 18 3 x 24 3 x 12 3x 12 3x 24 3 3 3 3 x 4 x 8 The solution set is x x 4 or x 8 The number line graph looks like Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16