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x3 9 Solve Absolute Value Inequalities 2x4 8 © 2011 The Enlightened Elephant Quick Intro Video https://www.youtube.com/watch?v=XCeH QgXRzFU First, let’s think about what absolute value inequalities are really asking us to find! What does x 4 x 4 All the numbers whose distance from zero is greater than 4. 4 -4 x 4 really mean? or x4 ***Notice that you need to have two inequalities to represent the distances that are greater than 4 from zero. What does x 4 x 4 really mean? All the numbers whose distance from zero is less than 4. x 4 -4 and 4 x4 ***Notice that you need to have two inequalities to represent the distances that are less than 4 from zero. However, since these inequalities must happen at the same time, it should be written as 4 x 4 OK, so how do we solve more difficult problems? Solve and graph 2 x 4 8 Step 1: Isolate the absolute value. x4 4 Step 2: Set up two inequalities. x4 4 x 4 4 Step 3: Solve the inequalities. x 8 x0 Step 4: Graph the solutions. 0 8 Let’s practice! You Try! Step 1: Solve and graph. 4 x 2 Step 2: Step 3: Step 4: 1 11 x2 3 x23 x 2 3 x 1 x 5 Step 5: -5 Final answer: -5<x<1 1 Set up two inequalities! You Try! x3 9 x 3 9 x 3 9 Solve and graph. x6 x 12 -12 Final answer: 6 x6 or x 12 Isolate the absolute value first! You Try! Solve and graph. 3 2 x 1 21 2x 1 7 2 x 1 7 x4 2x 1 7 x 3 -3 Final answer: 4 x4 or x 3 You Try! Solve and graph. Isolate the absolute value first! 3x 6 3 15 3x 6 12 3x 6 12 3x 6 12 x6 -2 Final answer: x 2 6 2 x 6 Isolate the absolute value first! You Try! Solve and graph. 3 2 x 1 8 23 2x 1 5 2x 1 5 x2 -3 Final answer: 2x 1 5 x 3 2 3 x 2 You Try! Solve and graph. Isolate the absolute value first! 2 x 8 12 Final answer: NO SOLUTION! Absolute values are distances, which cannot be negative. You can choose any value for x and the left side of the inequality will always be positive. However, positive numbers are NEVER less than negative numbers. You Try! Solve and graph. Isolate the absolute value first! 6x 12 10 Final answer: ALL REAL NUMBERS! Absolute values are distances, which cannot be negative. You can choose any value for x and the left side of the inequality will always be positive. Positive numbers are ALWAYS greater than negative numbers. Success! Nice job!