Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = 1, - 2

x=1 , -2

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| x + 8 | = | 5 x + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x + 8 \| = \| 5 x + 4 \|$
$x = + y$	$(x + 8) = (5 x + 4)$
$x = - y$	$(x + 8) = - (5 x + 4)$
$+ x = y$	$(x + 8) = (5 x + 4)$
$- x = y$	$- (x + 8) = (5 x + 4)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| x + 8 \| = \| 5 x + 4 \|$
$x = + y$ , $+ x = y$	$(x + 8) = (5 x + 4)$
$x = - y$ , $- x = y$	$(x + 8) = - (5 x + 4)$

2. Solve the two equations for x

12 additional steps

$(x + 8) = (5 x + 4)$

Subtract from both sides:

$(x + 8) - 5 x = (5 x + 4) - 5 x$

Group like terms:

$(x - 5 x) + 8 = (5 x + 4) - 5 x$

Simplify the arithmetic:

$- 4 x + 8 = (5 x + 4) - 5 x$

Group like terms:

$- 4 x + 8 = (5 x - 5 x) + 4$

Simplify the arithmetic:

$- 4 x + 8 = 4$

Subtract from both sides:

$(- 4 x + 8) - 8 = 4 - 8$

Simplify the arithmetic:

$- 4 x = 4 - 8$

Simplify the arithmetic:

$- 4 x = - 4$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 4}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 4}{- 4}$

Simplify the fraction:

$x = \frac{- 4}{- 4}$

Cancel out the negatives:

$x = \frac{4}{4}$

Simplify the fraction:

$x = 1$

12 additional steps

$(x + 8) = - (5 x + 4)$

Expand the parentheses:

$(x + 8) = - 5 x - 4$

Add to both sides:

$(x + 8) + 5 x = (- 5 x - 4) + 5 x$

Group like terms:

$(x + 5 x) + 8 = (- 5 x - 4) + 5 x$

Simplify the arithmetic:

$6 x + 8 = (- 5 x - 4) + 5 x$

Group like terms:

$6 x + 8 = (- 5 x + 5 x) - 4$

Simplify the arithmetic:

$6 x + 8 = - 4$

Subtract from both sides:

$(6 x + 8) - 8 = - 4 - 8$

Simplify the arithmetic:

$6 x = - 4 - 8$

Simplify the arithmetic:

$6 x = - 12$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{- 12}{6}$

Simplify the fraction:

$x = \frac{- 12}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 2 \cdot 6)}{(1 \cdot 6)}$

Factor out and cancel the greatest common factor:

$x = - 2$

3. List the solutions

$x = 1, - 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x + 8 |$
$y = | 5 x + 4 |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved