Absolute © 2009 by S - Squared, Inc. All Rights Reserved. Value

Absolute Values:  The absolute value of a number is the distance a number is from zero.  A distance is always positive.  An absolute value symbol is denoted with two vertical bars. │ │  The absolute value operation is a grouping symbol, so expressions should be simplified prior to applying the absolute value operation.

Absolute Values: Simplify: │3│ │3│ = 3  The absolute value of a number is the distance a number is from zero.

Absolute Values: Simplify: │− 3│ │− 3│ = 3  The absolute value of a number is a distance. Distances are always positive.

Absolute Values: Simplify: −│5 – 7│ −│− 2│ =  Simplify the numerical expression before applying the absolute value operation.  Apply the absolute value operation prior to any other operations outside the absolute value bars. − ( 2 ) = − 2 = Opposite of 2

Simplify: 7 + 6│15 – 5│ Absolute Values: 7 + 6│10│ = ( 10 ) = = Absolute Value Simplify Multiply 67 = Add

Absolute Values: Solve: │x│ = 5  When considering an absolute value operation in an equation, think of the possible inputs for x. 5 x 5 │− 5│ = and │5│ =  x has two possible values. x = − 5 and x = 5

Absolute Values: Solve: │2x – 9│ – 5 = 2  Isolate the absolute value operation. │2x – 9│ = 7 Add 5  2x – 9 has two possible values. 2x – 9 = 7 2x – 9 = −  Solve to find two possible values of x x = 16 2x = x = 8 and x = 1 Add 9 Divide by 2

Absolute Values:  Check the two solutions. x = 1, x = 8 Solutions Equation │2x – 9│ – 5 = 2 Substitute x = 1 │2(1) – 9│ – 5 = 2 Multiply │2 – 9│ – 5 = 2 Subtract │− 7 │ – 5 = 2 Absolute value 7 – 5 = 2 Subtract 2 = 2 Check confirms accuracy Substitute x = 8 │2(8) – 9│ – 5 = 2 Multiply │16 – 9│ – 5 = 2 Subtract │7 │ – 5 = 2 Absolute value 7 – 5 = 2 Subtract 2 = 2 Both solutions check

Absolute Values: Solve: │3x + 8│ + 11 = 6  Isolate the absolute value operation. │3x + 8│ = − 5 Subtract 11  Notice, we have an absolute value equal to a negative number. A distance can never be negative. There is no value of x that can make the equation true. No solution

Absolute Values: Graph: │2x│ > 2  When considering an absolute value inequality, start by writing the two related inequalities.  1 st inequality: Drop the absolute value bars. 2x > 2  2 nd inequality: Drop the absolute value bars, change direction of inequality and take the opposite of the number. 2x < − 2

Absolute Values: (Continued) Graph: │2x│ > 2  Solve the two related inequalities to graph. 2x > 2 2x < − 2 Divide by x > 1 x < − 1 Or x Graph

Absolute Values: Graph: │w + 3│ ≤ 2  When considering an absolute value inequality, start by writing the two related inequalities.  1 st inequality: Drop the absolute value bars. w + 3 ≤ 2  2 nd inequality: Drop the absolute value bars, change direction of inequality and take the opposite of the number. w + 3 ≥ − 2

Absolute Values: (Continued) Graph: │w + 3│ ≤ 2  Solve the two related inequalities to graph. w + 3 ≤ 2 w + 3 ≥ − 2 Subtract 3 w ≤ − 1 w ≥ − 5 And w Graph – 3 – 3– 3– 3– 3 – 3– 3– 3– 3 − 5 ≤ w ≤ − 1 − 5 ≤ w ≤ − 1 Other form for solution

Absolute Values: Summary  The absolute value is measuring the distance a number or expression is from zero. number or expression is from zero.  To solve an absolute value equation: * Isolate the absolute value operation. * Write two equations based on possible outcomes. * Solve the equations.  To solve and graph an absolute value inequality: * Isolate the absolute value operation. * Write two related inequalities * Solve the inequalities and graph.