Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = - \frac{29}{3}, \frac{3}{13}

x=-\frac{29}{3} , \frac{3}{13}

Mixed number form:

x = - 9 \frac{2}{3}, \frac{3}{13}

x=-9\frac{2}{3} , \frac{3}{13}

Decimal form:

x = - 9.667, 0.231

x=-9.667 , 0.231

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
$| 5 x - 16 | = | 8 x + 13 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x - 16 \| = \| 8 x + 13 \|$
$x = + y$	$(5 x - 16) = (8 x + 13)$
$x = - y$	$(5 x - 16) = - (8 x + 13)$
$+ x = y$	$(5 x - 16) = (8 x + 13)$
$- x = y$	$- (5 x - 16) = (8 x + 13)$

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:

$\| x \| = \| y \|$	$\| 5 x - 16 \| = \| 8 x + 13 \|$
$x = + y$ , $+ x = y$	$(5 x - 16) = (8 x + 13)$
$x = - y$ , $- x = y$	$(5 x - 16) = - (8 x + 13)$

2. Solve the two equations for x

11 additional steps

$(5 x - 16) = (8 x + 13)$

Subtract from both sides:

$(5 x - 16) - 8 x = (8 x + 13) - 8 x$

Group like terms:

$(5 x - 8 x) - 16 = (8 x + 13) - 8 x$

Simplify the arithmetic:

$- 3 x - 16 = (8 x + 13) - 8 x$

Group like terms:

$- 3 x - 16 = (8 x - 8 x) + 13$

Simplify the arithmetic:

$- 3 x - 16 = 13$

Add to both sides:

$(- 3 x - 16) + 16 = 13 + 16$

Simplify the arithmetic:

$- 3 x = 13 + 16$

Simplify the arithmetic:

$- 3 x = 29$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{29}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{29}{- 3}$

Simplify the fraction:

$x = \frac{29}{- 3}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 29}{3}$

10 additional steps

$(5 x - 16) = - (8 x + 13)$

Expand the parentheses:

$(5 x - 16) = - 8 x - 13$

Add to both sides:

$(5 x - 16) + 8 x = (- 8 x - 13) + 8 x$

Group like terms:

$(5 x + 8 x) - 16 = (- 8 x - 13) + 8 x$

Simplify the arithmetic:

$13 x - 16 = (- 8 x - 13) + 8 x$

Group like terms:

$13 x - 16 = (- 8 x + 8 x) - 13$

Simplify the arithmetic:

$13 x - 16 = - 13$

Add to both sides:

$(13 x - 16) + 16 = - 13 + 16$

Simplify the arithmetic:

$13 x = - 13 + 16$

Simplify the arithmetic:

$13 x = 3$

Divide both sides by :

$\frac{(13 x)}{13} = \frac{3}{13}$

Simplify the fraction:

$x = \frac{3}{13}$

3. List the solutions

$x = - \frac{29}{3}, \frac{3}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 x - 16 |$
$y = | 8 x + 13 |$
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