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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