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1.4 – Solving Absolute Value Equations

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Absolute Value

Absolute Value–unit value only

Absolute Value–unit value only

Absolute Value–unit value only (w/o signs)

Absolute Value–unit value only (w/o signs)

ex. |-5|

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| =

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 +

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3)

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4 +

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4 + |-15

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4 + |-15 – 7|

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4 + |-15 – 7|

=1.4

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4 + |-15 – 7|

=1.4 +

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4 + |-15 – 7|

=1.4 + |-22|

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4 + |-15 – 7|

=1.4 + |-22|

=1.4

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4 + |-15 – 7|

=1.4 + |-22|

=1.4 +

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4 + |-15 – 7|

=1.4 + |-22|

=1.4 + 22

Absolute Value–unit value only (w/o signs)

ex. |-5| = 5; |5| = 5

Example 1

Evaluate 1.4+|5y – 7| if y=-3

1.4+|5y – 7|=1.4 + |5(-3) – 7|

=1.4 + |-15 – 7|

=1.4 + |-22|

=1.4 + 22

= 23.4

Example 2

Example 2 Solve |x – 18| = 5.

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18

x = 23

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

|5x – 6| + 9 = 0

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

|5x – 6| + 9 = 0

-9 -9

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

|5x – 6| + 9 = 0

-9 -9

|5x – 6| = -9

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

|5x – 6| + 9 = 0

-9 -9

|5x – 6| = -9

Note:

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

|5x – 6| + 9 = 0

-9 -9

|5x – 6| = -9

Note: Absolute value cannot equal a negative number!

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

|5x – 6| + 9 = 0

-9 -9

|5x – 6| = -9

Note: Absolute value cannot equal a negative number!

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

|5x – 6| + 9 = 0

-9 -9

|5x – 6| = -9

Note: Absolute value cannot equal a negative number!

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

|5x – 6| + 9 = 0

-9 -9

|5x – 6| = -9

Note: Absolute value cannot equal a negative number!

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

|5x – 6| + 9 = 0

-9 -9

|5x – 6| = -9

Note: Absolute value cannot equal a negative number!

Example 2 Solve |x – 18| = 5.

|x – 18| = 5

x – 18 = 5x – 18 = -5

+18 +18 +18 +18

x = 23x = 13

Example 3 Solve |5x – 6| + 9 = 0.

|5x – 6| + 9 = 0

-9 -9

|5x – 6| = -9

Note: Absolute value cannot equal a negative number!

x = Ø

Example 4

Example 4 Solve |x + 6| = 3x – 2.

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 =

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x –

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x

-2x + 6 = -2

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x

-2x + 6 = -2

- 6 -6

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x

-2x + 6 = -2

- 6 -6

-2x

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x

-2x + 6 = -2

- 6 -6

-2x = -8

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x

-2x + 6 = -2

- 6 -6

-2x = -8

-2 -2

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x

-2x + 6 = -2

- 6 -6

-2x = -8

-2 -2

x = 4

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -2

- 6 -6

-2x = -8

-2 -2

x = 4

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x

- 6 -6

-2x = -8

-2 -2

x = 4

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6

- 6 -6

-2x = -8

-2 -2

x = 4

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6 = 2

- 6 -6

-2x = -8

-2 -2

x = 4

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6 = 2

- 6 -6 - 6 -6

-2x = -8

-2 -2

x = 4

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6 = 2

- 6 -6 - 6 -6

-2x = -84x = -4

-2 -2

x = 4

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6 = 2

- 6 -6 - 6 -6

-2x = -84x = -4

-2 -2 4 4

x = 4

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6 = 2

- 6 -6 - 6 -6

-2x = -84x = -4

-2 -2 4 4

x = 4 OR x = -1

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6 = 2

- 6 -6 - 6 -6

-2x = -84x = -4

-2 -2 4 4

x = 4 OR x = -1

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6 = 2

- 6 -6 - 6 -6

-2x = -84x = -4

-2 -2 4 4

x = 4 OR x = -1

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6 = 2

- 6 -6 - 6 -6

-2x = -84x = -4

-2 -2 4 4

x = 4 OR x = -1

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6 = 2

- 6 -6 - 6 -6

-2x = -84x = -4

-2 -2 4 4

x = 4 OR x = -1

Example 4 Solve |x + 6| = 3x – 2.

|x + 6| = 3x – 2

x + 6 = 3x – 2 x + 6 = -(3x – 2)

x + 6 = -3x – (-2)

x + 6 = 3x – 2 x + 6 = -3x + 2

-3x -3x+3x +3x

-2x + 6 = -24x + 6 = 2

- 6 -6 - 6 -6

-2x = -84x = -4

-2 -2 4 4

x = 4 OR x = -1