ABSOLUTE VALUE EQUATIONS AND INEQUALITIES

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESUNIT 01 LESSON 05

OBJECTIVESSTUDENTS WILL BE ABLE TO: Solve absolute value equations Transform an absolute value inequality into a compoundinequality Solve absolute value inequality.KEY VOCABULARY: Equation. Inequality. Absolute value.

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESAbsolute Value means . only how far a number is from zero:66-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10"6" is 6 away from zero,and " 6" is also 6 away from zero.So the absolute value of 6 is 6,and the absolute value of 6 is also 602

03ABSOLUTE VALUE EQUATIONS AND INEQUALITIESTo solve an absolute value equation, isolate the absolute value on oneside of the equal sign, and establish two cases:Case 1:Case 2: a ba bSet the expression inside the absolutevalue symbol equal to the other givenexpression. a ba -bSet the expression inside the absolutevalue symbol equal to the negation ofthe other given expression

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESREMEMBER!Absolute value is always positive (or zero). An equation such as x – 3 -5is never true. It has NO solution. The answer is the empty set 04

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESNOTICE!Always CHECK your answers. The two cases create "derived"equations. These derived equations may not always be true equivalents tothe original equation. Consequently, the roots of the derived equationsMUST BE CHECKED in the original equation so that you do not listextraneous roots as answers.05

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESPROBLEM 1Solve x-10 6SolutionCase 1Case 2x - 10 6x – 10 -6x 16x 406

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESPROBLEM 2Solve2 3x-3 12SolutionFirst simplify the equation to get the absolute in one side.Divide both sides by 2 3x – 3 607

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESPROBLEM 2Case 1Case 23x – 3 63x – 3 -63x 93x -6 3x 33x -3x -108

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESSolving an absolute value inequality problem is similar tosolving an absolute value equation.Start by isolating the absolute value on one side of theinequality symbol, then follow the rules below:09

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESIf the symbol is 0𝑟 If 𝑎 0, then the solutions to 𝑥 𝑎 are 𝑥 𝑎 or 𝑥 𝑎.(or) if 𝑎 0, all real numbers will satisfy 𝑥 𝑎think about it: absolute value is always positive (or zero), so , of course, itis greater than any negative number.10

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESIf the symbol is (0r ) If 𝑎 0, then the solutions to 𝑥 𝑎 and 𝑥 𝑎Also written as : 𝑎 𝑥 a(and) if 𝑎 0, there is no solution to 𝑥 𝑎think about it: absolute value is always positive (or zero), so , of course, itcannot be less than a negative number.11

12ABSOLUTE VALUE EQUATIONS AND INEQUALITIESPROBLEM 3Solve x – 20 5SolutionNote that there are two parts to the solution and that theconnecting word is "or".Case 1Case 2x – 20 5x – 20 -5x 25x 15So x 15 or x 2510010203040

ABSOLUTE VALUE EQUATIONS AND INEQUALITIESPROBLEM 4Solve 5 x 10 5SolutionWe need to isolate the absolute value on one side of the equation. x 10 1Note that there are two parts to the solution and that theconnecting word is "and".13

14ABSOLUTE VALUE EQUATIONS AND INEQUALITIESCase 1x 10 1x -9Case 2x 10 -1x -11So x -11 and x -9Also can be written as-11 x -9-11-10.5-10.0-9.5-9.0

ABSOLUTE VALUE EQUATIONS AND INEQUALITIES 05 NOTICE! Always CHECK your answers. The two cases create "derived" equations. These derived equations may not always be true equivalents to the original equation. Consequently, the roots of the derived equations MUST BE CHECKED in the original equation so that you do not list extraneous roots as answers.