AlgebraAlgebra

6.4 Solving Absolute Value 6.4 Solving Absolute Value Equations and InequalitiesEquations and Inequalities

Steps to Solve an Absolute Steps to Solve an Absolute Value EquationValue Equation

1)1) Isolate the absolute value expressionIsolate the absolute value expression

2)2) Drop the absolute value and create a Drop the absolute value and create a positive and a negative equationpositive and a negative equation

3)3) Solve eachSolve each

Solving an Absolute Value Solving an Absolute Value EquationEquation

►Solve lx – 4l = 8Solve lx – 4l = 8

x – 4 = 8 or x – 4 = -8x – 4 = 8 or x – 4 = -8

+4 +4 +4 +4+4 +4 +4 +4

x = 12 or x = -4x = 12 or x = -4

Mentally check your answers.Mentally check your answers.

Solving an Absolute Value Solving an Absolute Value EquationEquation

► Solve l5x + 1l + 3 = 14Solve l5x + 1l + 3 = 14Isolate the absolute value expression.Isolate the absolute value expression.l5x + 1l = 11l5x + 1l = 115x + 1 = 11 or 5x + 1 = -115x + 1 = 11 or 5x + 1 = -11 -1 -1 -1 -1-1 -1 -1 -1 5x = 10 or 5x = -125x = 10 or 5x = -12

x = 2 or x = - 2 and 2/5x = 2 or x = - 2 and 2/5

Mentally check your answers.Mentally check your answers.

-3-3

5 5

Investigating Absolute-Value Investigating Absolute-Value InequalitiesInequalities

See if each number is a solution as it appears. If it is, fill it in.See if each number is a solution as it appears. If it is, fill it in.

1) lxl ≤ 2 1) lxl ≤ 2 (The distance from 0 is less than or equal to (The distance from 0 is less than or equal to

2)2)

2) lxl > 32) lxl > 3(The distance from 0 is greater than 3)(The distance from 0 is greater than 3)

3) lx + 2l ≥ 13) lx + 2l ≥ 1

4) lx – 3l ≤ 24) lx – 3l ≤ 2

-2 10 2 3 4 5-1-3-4-5 -2 10 2 3 4 5-1-3-4-5

-2 10 2 3 4 5-1-3-4-5 -2 10 2 3 4 5-1-3-4

. . ...

. .... . . ...

. . ...

x ≤ 2 and x ≥ -2

.. .

x > 3 or x < -3

. . ...

or

6

and

Absolute Value < or ≤ is an and inequality Absolute Value > or ≥ is an or inequality

WHY?

Steps to Solve an Absolute Steps to Solve an Absolute Value InequalityValue Inequality

1)1) Isolate the Absolute Value ExpressionIsolate the Absolute Value Expression

2)2) Drop the absolute value and create a Drop the absolute value and create a positive inequality and a negative positive inequality and a negative inequality (flip the sign)inequality (flip the sign)

3)3) Solve each inequality separately Solve each inequality separately

4)4) <, ≤ is an <, ≤ is an andand inequality inequality

>, ≥ is an >, ≥ is an oror inequality inequality

5)5) Graph Graph

Let’s Try! Solve lx + 6l < 8Let’s Try! Solve lx + 6l < 8

x + 6 < 8 and x + 6 > -8x + 6 < 8 and x + 6 > -8

-6 -6 -6 -6-6 -6 -6 -6

x < 2 and x > -14x < 2 and x > -14

-14 0 2

You Try! Solve l2x - 4l ≥ 2You Try! Solve l2x - 4l ≥ 2

2x - 4 ≥ 2 or 2x - 4 ≤ -22x - 4 ≥ 2 or 2x - 4 ≤ -2

+4 +4 +4 +4+4 +4 +4 +4

2x ≥ 6 or 2x ≤ 22x ≥ 6 or 2x ≤ 2

x ≥ 3 or x ≤ 1x ≥ 3 or x ≤ 1

0 1 32

Let’s Try! Solve l2x + 1l - 3 ≥ Let’s Try! Solve l2x + 1l - 3 ≥ 66

l2x + 1l – 3 ≥ 6l2x + 1l – 3 ≥ 6

+3 +3+3 +3

l2x + 1l ≥ 9 l2x + 1l ≥ 9

2x + 1 ≥ 9 or 2x + 1 ≤ -92x + 1 ≥ 9 or 2x + 1 ≤ -9

-1 -1 -1 -1-1 -1 -1 -1

2x ≥ 8 or 2x ≤ -102x ≥ 8 or 2x ≤ -10

x ≥ 4 or x ≤ -5x ≥ 4 or x ≤ -5

0 4-5

You Try! Solve l3x - 3l + 4 < You Try! Solve l3x - 3l + 4 < 1010

l3x - 3l + 4 < 10l3x - 3l + 4 < 10

-4 -4-4 -4

l3x - 3l < 6 l3x - 3l < 6

3x - 3 < 6 and 3x - 3 > -63x - 3 < 6 and 3x - 3 > -6

+3 +3 +3 +3+3 +3 +3 +3

3x < 9 and 3x > -33x < 9 and 3x > -3

x < 3 and x > -1x < 3 and x > -1

0 3-1

HWHW

►P. 356-358 (19-59 Odd, 77-83 Odd)P. 356-358 (19-59 Odd, 77-83 Odd)