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| x | = 5 Absolute Value Equations x = 5 x = – 5 SameOpposite | 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.
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Copyright © 2011 Pearson Education, Inc.
Equations Involving Absolute Value8.28.21. Solve equations involving absolute value.
| 3 | =
| 0 | =| 2.5 | =
| – 7 | =
| – 4.8 | =
3
7
4.8
02.5
True or False?The absolute value of a number is always positive.
False
The absolute value of a number is The absolute value of a number is either positive or 0.either positive or 0.The absolute value of a number is The absolute value of a number is non-negativenon-negative..
Absolute Value
> 0
≥ 0Non-negative
| x | = 5
Absolute Value EquationsAbsolute Value Equations
x = 5 x = – 5Same Opposite
| x | = –2
No Solution
Two Solutions
Absolute Value PropertyIf |x| = a, where x is a variable or an expression and a 0, then x = a or x = a.
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.
Absolute Value EquationsAbsolute Value Equations
9523 x
423 x
55
Same Opposite
423 xDrop the absolute value bars!
Keep the absolute value bars!
423 x33 12 x22
21x
33
22 72 x
27x
27
21 ,
1. Isolate2. Two Cases3. Solve
Absolute Value EquationsAbsolute Value Equations
6413 k
1013 k
44
Same Opposite
1013 kDrop the absolute value bars!
Keep the absolute value bars!
1013 k11 113 k33
311
k
11
3393 k
3k
3113,
1. Isolate2. Two Cases3. Solve
Absolute Value EquationsAbsolute Value Equations
7127 x
57 x
1212
1. Isolate2. Two Cases3. Solve
No Solution
Absolute Value EquationsAbsolute Value Equations
052 y
Same Opposite
052 y55 52 y22
25y
25
1. Isolate2. Two Cases3. Solve
Opposite
Absolute Value Equations with 2 Absolute ValuesAbsolute Value Equations with 2 Absolute Values
| 5 | = | 5 |Same Opposite
| –8 | = | –8 |
| 5 | = | –5 || –8 | = | 8 |
4x 4 x 4 x7x 7 x 7 x
32 x 32 x 32 x
Two cases!
Absolute Value Equations with 2 Absolute ValuesAbsolute Value Equations with 2 Absolute Values
2342 ww
Same Opposite
2342 ww 2342 ww
6
52 ,
ww 33 24 w44
6 w
6w
2342 ww
245 w
ww 33
44 25 w55
52w
Absolute Value Equations with 2 Absolute ValuesAbsolute Value Equations with 2 Absolute Values
332 kk
Same Opposite
332 kk 332 kk
21
45 ,
kk 322 k33
k21
21k
332 kk
324 k
kk 33
22 54 k44
45
k
22
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 x99
124 x
3x
xx 2392
39
xx 22
44No Solution