Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = - \frac{5}{7}, \frac{7}{5}

x=-\frac{5}{7} , \frac{7}{5}

Mixed number form:

x = - \frac{5}{7}, 1 \frac{2}{5}

x=-\frac{5}{7} , 1\frac{2}{5}

Decimal form:

x = - 0.714, 1.4

x=-0.714 , 1.4

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
$| - 18 x + 3 | = | 3 x + 18 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 18 x + 3 \| = \| 3 x + 18 \|$
$x = + y$	$(- 18 x + 3) = (3 x + 18)$
$x = - y$	$(- 18 x + 3) = - (3 x + 18)$
$+ x = y$	$(- 18 x + 3) = (3 x + 18)$
$- x = y$	$- (- 18 x + 3) = (3 x + 18)$

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:

$\| x \| = \| y \|$	$\| - 18 x + 3 \| = \| 3 x + 18 \|$
$x = + y$ , $+ x = y$	$(- 18 x + 3) = (3 x + 18)$
$x = - y$ , $- x = y$	$(- 18 x + 3) = - (3 x + 18)$

2. Solve the two equations for x

13 additional steps

$(- 18 x + 3) = (3 x + 18)$

Subtract from both sides:

$(- 18 x + 3) - 3 x = (3 x + 18) - 3 x$

Group like terms:

$(- 18 x - 3 x) + 3 = (3 x + 18) - 3 x$

Simplify the arithmetic:

$- 21 x + 3 = (3 x + 18) - 3 x$

Group like terms:

$- 21 x + 3 = (3 x - 3 x) + 18$

Simplify the arithmetic:

$- 21 x + 3 = 18$

Subtract from both sides:

$(- 21 x + 3) - 3 = 18 - 3$

Simplify the arithmetic:

$- 21 x = 18 - 3$

Simplify the arithmetic:

$- 21 x = 15$

Divide both sides by :

$\frac{(- 21 x)}{- 21} = \frac{15}{- 21}$

Cancel out the negatives:

$\frac{21 x}{21} = \frac{15}{- 21}$

Simplify the fraction:

$x = \frac{15}{- 21}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 15}{21}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 3)}{(7 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 5}{7}$

14 additional steps

$(- 18 x + 3) = - (3 x + 18)$

Expand the parentheses:

$(- 18 x + 3) = - 3 x - 18$

Add to both sides:

$(- 18 x + 3) + 3 x = (- 3 x - 18) + 3 x$

Group like terms:

$(- 18 x + 3 x) + 3 = (- 3 x - 18) + 3 x$

Simplify the arithmetic:

$- 15 x + 3 = (- 3 x - 18) + 3 x$

Group like terms:

$- 15 x + 3 = (- 3 x + 3 x) - 18$

Simplify the arithmetic:

$- 15 x + 3 = - 18$

Subtract from both sides:

$(- 15 x + 3) - 3 = - 18 - 3$

Simplify the arithmetic:

$- 15 x = - 18 - 3$

Simplify the arithmetic:

$- 15 x = - 21$

Divide both sides by :

$\frac{(- 15 x)}{- 15} = \frac{- 21}{- 15}$

Cancel out the negatives:

$\frac{15 x}{15} = \frac{- 21}{- 15}$

Simplify the fraction:

$x = \frac{- 21}{- 15}$

Cancel out the negatives:

$x = \frac{21}{15}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(7 \cdot 3)}{(5 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{7}{5}$

3. List the solutions

$x = - \frac{5}{7}, \frac{7}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 18 x + 3 |$
$y = | 3 x + 18 |$
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