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Solve Absolute Value Inequalities When an inequality contains an absolute value, the absolute value must be removed in order to graph the solution and/or give interval notation. The way the absolute value is removed depends on the direction of the inequality symbol. Consider |x| < 2. Since absolute value is the distance from zero, this would be read as “the distance from zero is less than 2”. So on a number line, the solution area is all points less than 2 units away from zero. -2 < x < 2 Notice that the graph looks just like a conjunction. The interval notation is (-2, 2). Consider |x| > 2 Since absolute value is the distance from zero, this would be read as “the distance from zero is greater than 2”. So on a number line, the solution area is all points more than 2 units away from zero. x < -2 or x > 2 Notice that the graph looks just like a disjunction. The interval notation is (-∞, -2) ∪ (2, ∞). Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) To solve absolute value inequalities, the procedure is similar to that used when solving absolute value equations: Isolate the absolute value. If the inequality is greater than (>) or greater than or equal to (>), solve the separate inequalities. If the inequality is less than (<) or less than or equal to (<), make a three-part inequality (tripartite), and solve. Remember, when multiplying or dividing by a negative, the inequality symbol will switch directions. Example 1: Solve, graph, and write interval notation for the solution |4x − 5| > 6 > so use separate inequalities 4x – 5 < -6 +5 +5 4x < -1 4 4 x<- or 4x – 5 > 6 +5+5 4x > 11 4 4 1 x> 4 1 11 4 4 (−∞, − ] ∪ [ Solve each inequality Add 5 to all sides Divide all sides by 4 11 4 , ∞) Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Example 2: Solve, graph, and write interval notation for the solution -4 - 3|x| < -16 Isolate the absolute value +4 +4 Add 4 to both sides - 3|x| < -12 Divide both sides by -3 -3 -3 Switch the inequality sign |x| > 4 > so use separate inequalities x < -4 or x > 4 (-∞, -4] ∪ [4, ∞) Example 3: Solve, graph, and write interval notation for the solution 9 - 2|4x + 1| > 3 Isolate the absolute value -9 -9 Subtract 9 from both sides - 2|4x + 1| > -6 Divide both sides by -2 -2 -2 Switch the inequality sign |4x + 1| < 3 < so use the tripartite -3 < 4x + 1 < 3 -1 -1 -1 -4 < 4x < 2 4 4 4 -1 < x < Subtract 1 from all sides Divide all sides by 4 1 2 1 (−1, ) 2 Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) It is important to remember that when solving these equations, the absolute value is always positive – greater than a negative. An absolute value less than a negative number indicates no solution. Similarly, an absolute value greater than a negative indicates a solution of all real numbers. Example 4: Solve, graph, and write interval notation for the solution 12 + 4|6x − 1| < 4 Isolate the absolute value - 12 - 12 Subtract 12 from both sides 4|6x − 1| < -8 4 4 Divide both sides by 4 |6x − 1| < -2 Since absolute value cannot be less than a negative, there is no solution or ∅. Example 5: Solve, graph, and write interval notation for the solution 5 - 6|x + 7| < 17 Isolate the absolute value -5 -5 Subtract 5 from both sides -6|x + 7| < 12 -6 -6 Divide both sides by -6 |x + 7| > -2 Switch the inequality sign Since absolute value is always greater than a negative, the solution is all real numbers. (-∞, ∞) Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)