What do you do when you have absolute values on both sides of the equations?

Question

Hunter Terry · Accepted Answer

When both sides of the equations have absolute values,
we must consider both possibilities for acceptable solutions - positive and negative absolute value expressions.
To better comprehend, let's first look at an examle:
Example-1
Solve for

x

:

| 2 x - 1 | = | 4 x + 9 |

Both sides of the equation contain absolute values.
Find the answers as indicated below:

(2 x - 1) = - (4 x + 9)

.. Exp.1

O R

(2 x - 1) = (4 x + 9)

...Exp.2

C a s e .1

:
Consider ... Exp.1 first and solve for

x

(2 x - 1) = - (4 x + 9)

\Rightarrow 2 x - 1 = - 4 x - 9

Add

4 x

to both sides of the equation.

\Rightarrow 2 x - 1 + 4 x = - 4 x - 9 + 4 x

\Rightarrow 2 x - 1 + 4 x = - 4 x - 9 + 4 x

\Rightarrow 6 x - 1 = - 9

Add

1

to both sides of the equation.

\Rightarrow 6 x - 1 + 1 = - 9 + 1

\Rightarrow 6 x - 1 + 1 = - 9 + 1

\Rightarrow 6 x = - 8

Divide both sides by

2

\Rightarrow \frac{6 x}{2} = - \frac{8}{2}

\Rightarrow 3 x = - 4

\Rightarrow x = (- \frac{4}{3})

... Sol.1

C a s e .2

:
Consider ... Exp.2 next and solve for

x

(2 x - 1) = (4 x + 9)

\Rightarrow 2 x - 1 = 4 x + 9

Subtract

(4 x)

from both sides of the equation.

\Rightarrow 2 x - 1 - 4 x = 4 x + 9 - 4 x

\Rightarrow 2 x - 1 - 4 x = 4 x + 9 - 4 x

\Rightarrow - 2 x - 1 = 9

Add

1

to both sdies of the equation.

\Rightarrow - 2 x - 1 + 1 = 9 + 1

\Rightarrow - 2 x - 1 + 1 = 9 + 1

\Rightarrow - 2 x = 10

Divide both sides of the equation by

2

\Rightarrow \frac{- 2 x}{2} = \frac{10}{2}

\Rightarrow - x = 5

\Rightarrow x = - 5

... Sol.2
Thus, there are two solutions for the absolute value equation:

\Rightarrow x = (- \frac{4}{3})

... Sol.1

\Rightarrow x = - 5

... Sol.2
If you so wish, you can substitute these values of

x

in both

C a s e .1

and

C a s e .2

to verify the accuracy.
Let us consider a second example:
Example.2
Solve for

x

:

5 | x + 3 | - 4 = 8 | x + 3 | - 4

Subtract

8 | x + 3 |

and add

4

on both sides:

\Rightarrow 5 | x + 3 | - 4 - 8 | x + 3 | + 4 = 8 | x + 3 | - 4 - 8 | x + 3 | + 4

\Rightarrow 5 | x + 3 | - 4 - 8 | x + 3 | + 4 = 8 | x + 3 | - 4 - 8 | x + 3 | + 4

\Rightarrow 5 | x + 3 | - 8 | x + 3 | = - 4 + 4

\Rightarrow - 3 | x + 3 | = 0

Divide both sides by

(- 3)

\Rightarrow \frac{(- 3) (| x + 3 |)}{(- 3)} = \frac{0}{(- 3)}

\Rightarrow \frac{- 3 (| x + 3 |)}{- 3} = 0

\Rightarrow | x + 3 | = 0

\Rightarrow x + 3 = 0

Subtract

3

from both sides

\Rightarrow x + 3 - 3 = 0 - 3

\Rightarrow x + 3 - 3 = - 3

\Rightarrow x = - 3

Hence, we conclude that

x = - 3

is the ONLY Solution for this example.

What do you do when you have absolute values on both sides of the equations?

Answered question

Answer & Explanation

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