algebra 6.4 solving absolute value equations and inequalities
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AlgebraAlgebra
6.4 Solving Absolute Value 6.4 Solving Absolute Value Equations and InequalitiesEquations and Inequalities
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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
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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.
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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.
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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?
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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
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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
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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
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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
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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
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HWHW
►P. 356-358 (19-59 Odd, 77-83 Odd)P. 356-358 (19-59 Odd, 77-83 Odd)