Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{13}{3}, - \frac{1}{6}

x=\frac{13}{3} , -\frac{1}{6}

Mixed number form:

x = 4 \frac{1}{3}, - \frac{1}{6}

x=4\frac{1}{3} , -\frac{1}{6}

Decimal form:

x = 4.333, - 0.167

x=4.333 , -0.167

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
$| 9 x - 12 | = | 3 x + 14 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 x - 12 \| = \| 3 x + 14 \|$
$x = + y$	$(9 x - 12) = (3 x + 14)$
$x = - y$	$(9 x - 12) = - (3 x + 14)$
$+ x = y$	$(9 x - 12) = (3 x + 14)$
$- x = y$	$- (9 x - 12) = (3 x + 14)$

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:

$\| x \| = \| y \|$	$\| 9 x - 12 \| = \| 3 x + 14 \|$
$x = + y$ , $+ x = y$	$(9 x - 12) = (3 x + 14)$
$x = - y$ , $- x = y$	$(9 x - 12) = - (3 x + 14)$

2. Solve the two equations for x

11 additional steps

$(9 x - 12) = (3 x + 14)$

Subtract from both sides:

$(9 x - 12) - 3 x = (3 x + 14) - 3 x$

Group like terms:

$(9 x - 3 x) - 12 = (3 x + 14) - 3 x$

Simplify the arithmetic:

$6 x - 12 = (3 x + 14) - 3 x$

Group like terms:

$6 x - 12 = (3 x - 3 x) + 14$

Simplify the arithmetic:

$6 x - 12 = 14$

Add to both sides:

$(6 x - 12) + 12 = 14 + 12$

Simplify the arithmetic:

$6 x = 14 + 12$

Simplify the arithmetic:

$6 x = 26$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{26}{6}$

Simplify the fraction:

$x = \frac{26}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(13 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

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

12 additional steps

$(9 x - 12) = - (3 x + 14)$

Expand the parentheses:

$(9 x - 12) = - 3 x - 14$

Add to both sides:

$(9 x - 12) + 3 x = (- 3 x - 14) + 3 x$

Group like terms:

$(9 x + 3 x) - 12 = (- 3 x - 14) + 3 x$

Simplify the arithmetic:

$12 x - 12 = (- 3 x - 14) + 3 x$

Group like terms:

$12 x - 12 = (- 3 x + 3 x) - 14$

Simplify the arithmetic:

$12 x - 12 = - 14$

Add to both sides:

$(12 x - 12) + 12 = - 14 + 12$

Simplify the arithmetic:

$12 x = - 14 + 12$

Simplify the arithmetic:

$12 x = - 2$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{- 2}{12}$

Simplify the fraction:

$x = \frac{- 2}{12}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{6}$

3. List the solutions

$x = \frac{13}{3}, - \frac{1}{6}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 x - 12 |$
$y = | 3 x + 14 |$
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