Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

w = \frac{13}{11}, - \frac{1}{13}

w=\frac{13}{11} , -\frac{1}{13}

Mixed number form:

w = 1 \frac{2}{11}, - \frac{1}{13}

w=1\frac{2}{11} , -\frac{1}{13}

Decimal form:

w = 1.182, - 0.077

w=1.182 , -0.077

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
$| 12 w - 6 | = | w + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 12 w - 6 \| = \| w + 7 \|$
$x = + y$	$(12 w - 6) = (w + 7)$
$x = - y$	$(12 w - 6) = - (w + 7)$
$+ x = y$	$(12 w - 6) = (w + 7)$
$- x = y$	$- (12 w - 6) = (w + 7)$

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:

$\| x \| = \| y \|$	$\| 12 w - 6 \| = \| w + 7 \|$
$x = + y$ , $+ x = y$	$(12 w - 6) = (w + 7)$
$x = - y$ , $- x = y$	$(12 w - 6) = - (w + 7)$

2. Solve the two equations for w

9 additional steps

$(12 w - 6) = (w + 7)$

Subtract from both sides:

$(12 w - 6) - w = (w + 7) - w$

Group like terms:

$(12 w - w) - 6 = (w + 7) - w$

Simplify the arithmetic:

$11 w - 6 = (w + 7) - w$

Group like terms:

$11 w - 6 = (w - w) + 7$

Simplify the arithmetic:

$11 w - 6 = 7$

Add to both sides:

$(11 w - 6) + 6 = 7 + 6$

Simplify the arithmetic:

$11 w = 7 + 6$

Simplify the arithmetic:

$11 w = 13$

Divide both sides by :

$\frac{(11 w)}{11} = \frac{13}{11}$

Simplify the fraction:

$w = \frac{13}{11}$

10 additional steps

$(12 w - 6) = - (w + 7)$

Expand the parentheses:

$(12 w - 6) = - w - 7$

Add to both sides:

$(12 w - 6) + w = (- w - 7) + w$

Group like terms:

$(12 w + w) - 6 = (- w - 7) + w$

Simplify the arithmetic:

$13 w - 6 = (- w - 7) + w$

Group like terms:

$13 w - 6 = (- w + w) - 7$

Simplify the arithmetic:

$13 w - 6 = - 7$

Add to both sides:

$(13 w - 6) + 6 = - 7 + 6$

Simplify the arithmetic:

$13 w = - 7 + 6$

Simplify the arithmetic:

$13 w = - 1$

Divide both sides by :

$\frac{(13 w)}{13} = \frac{- 1}{13}$

Simplify the fraction:

$w = \frac{- 1}{13}$

3. List the solutions

$w = \frac{13}{11}, - \frac{1}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 12 w - 6 |$
$y = | w + 7 |$
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 w

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 w

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved