1. The document discusses solving equations involving absolute value, including equations with a single absolute value and equations with two absolute values. 2. To solve an equation with a single absolute value, the equation is isolated so it is in the form |ax + b| = c, then the absolute value is separated into two cases: ax + b = c and ax + b = -c. These two equations are then solved. 3. To solve an equation with two absolute values, the equation is also separated into two cases since the values inside the absolute values could be the same or opposite. Each case is then solved.

1. Copyright © 2011 Pearson Education, Inc.
Equations Involving Absolute Value
8.28.2
1. Solve equations involving absolute value.

2. | 3 | =
| 0 | =
| 2.5 | =
| – 7 | =
| – 4.8 | =
3
7
4.8
0
2.5
True or False?
The absolute value of a number is always positive.
False
he absolute value of a number ishe absolute value of a number is either positive or 0.either positive or 0.
The absolute value of a number isThe absolute value of a number is non-negativenon-negative..
Absolute Value
> 0
≥ 0
Non-negative

3. | x | = 5
Absolute Value EquationsAbsolute Value Equations
x = 5 x = – 5
Same Opposite
| x | = –2
No Solution
Two Solutions
Absolute Value Property
If |x| = a, where x is a variable or an expression and
a ≥ 0, then x = a or x = −a.

4. SolvingSolving Absolute Value EquationsAbsolute Value Equations
1. Isolate the absolute value so that the equation is
in the form |ax + b| = c.
If c < 0, the equation has no solution.
2. Separate the absolute value into two equations,
ax + b = c and ax + b = − c.
3. Solve both equations.

5. Absolute Value EquationsAbsolute Value Equations
9523 =+− x
423 =− x
55 −−
Same Opposite
423 =− xDrop the
absolute
value bars!
Keep the
absolute
value bars!
423 −=− x
33 −−
12 =− x
22 −−
2
1−
=x
33 −−
22 −−
72 −=− x
2
7
=x





−
2
7
2
1
,
1. Isolate
2. Two Cases
3. Solve

6. Absolute Value EquationsAbsolute Value Equations
6413 =−−k
1013 =−k
44 ++
Same Opposite
1013 =−kDrop the
absolute
value bars!
Keep the
absolute
value bars!
1013 −=−k
11 ++
113 =k
33
3
11
=k
11 ++
33
93 −=k
3−=k





−
3
11
3,
1. Isolate
2. Two Cases
3. Solve

7. Absolute Value EquationsAbsolute Value Equations
7127 =++x
57 −=+x
1212 −−
1. Isolate
2. Two Cases
3. Solve
No Solution

8. Absolute Value EquationsAbsolute Value Equations
052 =+y
Same Opposite
052 =+y
55 −−
52 −=y
22
2
5−
=y





−
2
5
1. Isolate
2. Two Cases
3. Solve

9. Opposite
Absolute Value Equations with 2 Absolute ValuesAbsolute Value Equations with 2 Absolute Values
| 5 | = | 5 |
Same Opposite
| –8 | = | –8 |
| 5 | = | –5 |
| –8 | = | 8 |
4+x ( )4+− x 4−−= x
7−x ( )7−− x 7+−= x
32 +− x ( )32 +−− x 32 −= x
Two cases!

10. Absolute Value Equations with 2 Absolute ValuesAbsolute Value Equations with 2 Absolute Values
2342 −=+ ww
Same Opposite
2342 −=+ ww ( )2342 −−=+ ww





− 6
5
2
,
ww 33 −−
24 −=+− w
44 −−
6−=− w
6=w
2342 +−=+ ww
245 =+w
ww 33 ++
44 −−
25 −=w
55
5
2−
=w

11. Absolute Value Equations with 2 Absolute ValuesAbsolute Value Equations with 2 Absolute Values
332 +=+ kk
Same Opposite
332 +=+ kk ( )332 +−=+ kk





 −−
2
1
4
5
,
kk −−
322 += k
33 −−
k21 =−
2
1−
=k
332 −−=+ kk
324 −=+k
kk 33 ++
22 −−
54 −=k
44
4
5−
=k
22

12. Absolute Value Equations with 2 Absolute ValuesAbsolute Value Equations with 2 Absolute Values
xx 2392 −=−
Same Opposite
xx 2392 −=− ( )xx 2392 −−=−
{ }3
xx 22 ++
394 =−x
99 ++
124 =x
3=x
xx 2392 +−=−
39 −=−
xx 22 −−
44
No Solution