Transcript Algebra

Algebra
6.4 Solving Absolute Value
Equations and Inequalities
Steps to Solve an Absolute Value
Equation
1)
2)
3)
Isolate the absolute value expression
Drop the absolute value and create a
positive and a negative equation
Solve each
Solving an Absolute Value Equation
► Solve
lx – 4l = 8
x–4=8
+4 +4
x = 12
or
or
Mentally check your answers.
x – 4 = -8
+4 +4
x = -4
Solving an Absolute Value Equation
► Solve
l5x + 1l + 3 = 14
-3
-3
Isolate the absolute value expression.
l5x + 1l = 11
5x + 1 = 11
or
5x + 1 = -11
-1 -1
-1
-1
5x = 10
or
5x = -12
5
x=2
5
or
Mentally check your answers.
x = - 2 and 2/5
Investigating Absolute-Value Inequalities
See if each number is a solution as it appears. If it is, fill it in.
1) lxl ≤ 2
3) lx + 2l ≥ 1
(The distance from 0 is less than or equal to 2)
. . .
. . . . .
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
x ≤ 2 and x ≥ -2
or
2) lxl > 3
4) lx – 3l ≤ 2
(The distance from 0 is greater than 3)
. . ..
. . . . . . .
. . . . .
.. . .
-5 -4 -3 -2 -1 0 1 2 3 4 5
x > 3 or x < -3
-4 -3 -2 -1 0 1 2 3 4 5 6
WHY?
Absolute Value < or ≤ is an and inequality
and
Absolute Value > or ≥ is an or inequality
Steps to Solve an Absolute Value
Inequality
1)
2)
3)
4)
5)
Isolate the Absolute Value Expression
Drop the absolute value and create a
positive inequality and a negative
inequality (flip the sign)
Solve each inequality separately
<, ≤ is an and inequality
>, ≥ is an or inequality
Graph
Let’s Try! Solve lx + 6l < 8
x+6<8
-6 -6
x<2
-14
and
and
x + 6 > -8
-6 -6
x > -14
0
2
You Try! Solve l2x - 4l ≥ 2
2x - 4 ≥ 2
+4 +4
2x ≥ 6
x≥3
or
or
or
0
1
2
3
2x - 4 ≤ -2
+4 +4
2x ≤ 2
x≤1
Let’s Try! Solve l2x + 1l - 3 ≥ 6
l2x + 1l – 3 ≥ 6
+3 +3
l2x + 1l ≥ 9
2x + 1 ≥ 9
or
-1 -1
2x ≥ 8
or
x≥4
or
-5
0
2x + 1 ≤ -9
-1
-1
2x ≤ -10
x ≤ -5
4
You Try! Solve l3x - 3l + 4 < 10
l3x - 3l + 4 < 10
-4
-4
l3x - 3l < 6
3x - 3 < 6
and
+3 +3
3x < 9
and
x<3
and
-1
0
3x - 3 > -6
+3
+3
3x > -3
x > -1
3
HW
► P.
356-358 (19-59 Odd, 77-83 Odd)