1.6 Absolute Value 1.6 Absolute Value Inequalities and EquationsInequalities and Equations

Honors Algebra IIHonors Algebra II

Absolute ValueAbsolute ValueDistance from zero on the number line.Distance from zero on the number line.

Absolute ValueAbsolute ValueDistance from zero on the number line.Distance from zero on the number line.Always positive result for absolute value.Always positive result for absolute value.

Absolute ValueAbsolute ValueDistance from zero on the number line.Distance from zero on the number line.Always positive result for absolute value.Always positive result for absolute value.Works as a grouping in order of Works as a grouping in order of

operations, but it is NOT parentheses operations, but it is NOT parentheses

=> don’t distribute!=> don’t distribute!

Absolute ValueAbsolute Value

|x| = x if x |x| = x if x ≥≥ 0 0 |x| = -x if x < 0|x| = -x if x < 0

Steps to Solve Absolute Value EquationsSteps to Solve Absolute Value Equations

Isolate absolute value expressionIsolate absolute value expressionSplit into two equationsSplit into two equationsSolve bothSolve bothCheck both Check both

Absolute Value EquationsAbsolute Value Equations

Solve:Solve: |2x – 1| = 5 |2x – 1| = 5

Absolute Value EquationsAbsolute Value Equations

CHECK:CHECK: |2x – 1| = 5 |2x – 1| = 5

x = 3x = 3 x = -2x = -2

Absolute Value EquationsAbsolute Value Equations

Solve:Solve: 3 |4w – 1| - 5 = 10 3 |4w – 1| - 5 = 10

Absolute Value EquationsAbsolute Value Equations

Check:Check: 3 |4w – 1| - 5 = 10 3 |4w – 1| - 5 = 10

Absolute Value EquationsAbsolute Value Equations

Solve:Solve: |2x + 5| = 3x + 4 |2x + 5| = 3x + 4

Absolute Value EquationsAbsolute Value Equations

Check:Check: |2x + 5| = 3x + 4 |2x + 5| = 3x + 4

Extraneous SolutionsExtraneous SolutionsSolutions that, though solved for correctly, Solutions that, though solved for correctly,

do not work in the original equation.do not work in the original equation.YOU MUST CHECK YOUR ANSWERS!YOU MUST CHECK YOUR ANSWERS!

Absolute Value EquationsAbsolute Value Equations

Absolute Value InequalitiesAbsolute Value Inequalities

Absolute Value InequalitiesAbsolute Value Inequalities

Steps to Solve Absolute Value InequalitiesSteps to Solve Absolute Value Inequalities

Isolate absolute value expressionIsolate absolute value expressionSplit into two inequalitiesSplit into two inequalitiesSolve bothSolve bothCheckCheck

Solve:Solve: |3x + 6||3x + 6| ≥ 12 ≥ 12

Absolute Value InequalitiesAbsolute Value Inequalities

Solve:Solve: 3 |2x + 6| - 9 < 153 |2x + 6| - 9 < 15

Absolute Value InequalitiesAbsolute Value Inequalities

SummarySummary Isolate Absolute ValueIsolate Absolute Value If =, OR problem If =, OR problem |x| = 5|x| = 5

x = 5 or x = -5x = 5 or x = -5 If >, If >, ≥, OR problem ≥, OR problem |x| > 5|x| > 5

x < -5 or x > 5x < -5 or x > 5 If <, If <, ≤, AND problem≤, AND problem |x| < 5|x| < 5

-5 < x < 5-5 < x < 5

x > -5 and x < 5x > -5 and x < 5

Tolerance: one half the difference of the Tolerance: one half the difference of the maximum and minimum acceptable maximum and minimum acceptable values. values.

Manufacturing ApplicationManufacturing Application

More Examples:More Examples:1.) 2 |3x – 1| + 5 = 331.) 2 |3x – 1| + 5 = 33

2.) |5z + 3| - 7 < 342.) |5z + 3| - 7 < 34

3.) -3 |2x – 3| + 1 3.) -3 |2x – 3| + 1 ≤ -20≤ -20