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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Absolute Value Equations

## Solving an Equation of the Form | z| = |w|

Here are several examples of solving equations of the form |z| = |w|.

Example 1

Solve: |4x - 34| = |6x + 14|

 Solution Replace 4x = 34 with z and 6x + 14 with w: So: |4x - 34| = |6x + 14| |z| = |w| z = w or z = -w Substitute 4x - 34 for z and 6x + 14 for w. Now, solve for x.     So: 4x - 34  4x -2x   x = 6x + 14= 6x + 48 = 48   = -24 or or or or or 4x - 34 =4x - 34 = 4x = 10x = x = -(6x + 14)-6x - 14 -6x + 20 20 2

Let's check the solutions:

 Check x = -24 Check x = -3 |4x - 34| Is |4(-24) - 34| Is |-130| Is 130 = |6x + 14| = |6(-24) + 14| ? = |-130| ? = 130 ? Yes |4x - 34| Is |4(2) - 34| Is |-26| Is 26 = |6x + 14|= |6(2) + 14| = |26| ? = 26 ? Yes

So, the solutions are x = -24 and x = 2.

Example 2

Solve: |3x - 4| = |3x + 16|

 Solution Replace 3x - 4 with z and 3x + 16 with w: So: |3x - 4| = |3x + 16||z| = |w| z = w or z = -w Substitute 3x - 4 for z and 3x + 16 for w. Now, solve for x. 3x + 4 3x 0 = 3x + 16= 3x + 20 = 20 or or or 3x - 4 = 3x - 4 = 6x = x = -(3x + 16)-3x - 16 -12 -2

Since 0 = 20 is a contradiction, the left equation does not lead to a solution.

 Check x = -2 |3x - 4| Is |3(-2) - 4| Is |-10| Is 10 = |3x + 16|= |3(-2) + 16| ? = |10| ? = 10 ? Yes

Thus, -2 is the only solution.

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