Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{6}{11}, \frac{4}{13}

x=\frac{6}{11} , \frac{4}{13}

Decimal form:

x = 0.545, 0.308

x=0.545 , 0.308

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
$| x + 1 | = | 12 x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x + 1 \| = \| 12 x - 5 \|$
$x = + y$	$(x + 1) = (12 x - 5)$
$x = - y$	$(x + 1) = - (12 x - 5)$
$+ x = y$	$(x + 1) = (12 x - 5)$
$- x = y$	$- (x + 1) = (12 x - 5)$

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:

$\| x \| = \| y \|$	$\| x + 1 \| = \| 12 x - 5 \|$
$x = + y$ , $+ x = y$	$(x + 1) = (12 x - 5)$
$x = - y$ , $- x = y$	$(x + 1) = - (12 x - 5)$

2. Solve the two equations for x

11 additional steps

$(x + 1) = (12 x - 5)$

Subtract from both sides:

$(x + 1) - 12 x = (12 x - 5) - 12 x$

Group like terms:

$(x - 12 x) + 1 = (12 x - 5) - 12 x$

Simplify the arithmetic:

$- 11 x + 1 = (12 x - 5) - 12 x$

Group like terms:

$- 11 x + 1 = (12 x - 12 x) - 5$

Simplify the arithmetic:

$- 11 x + 1 = - 5$

Subtract from both sides:

$(- 11 x + 1) - 1 = - 5 - 1$

Simplify the arithmetic:

$- 11 x = - 5 - 1$

Simplify the arithmetic:

$- 11 x = - 6$

Divide both sides by :

$\frac{(- 11 x)}{- 11} = \frac{- 6}{- 11}$

Cancel out the negatives:

$\frac{11 x}{11} = \frac{- 6}{- 11}$

Simplify the fraction:

$x = \frac{- 6}{- 11}$

Cancel out the negatives:

$x = \frac{6}{11}$

10 additional steps

$(x + 1) = - (12 x - 5)$

Expand the parentheses:

$(x + 1) = - 12 x + 5$

Add to both sides:

$(x + 1) + 12 x = (- 12 x + 5) + 12 x$

Group like terms:

$(x + 12 x) + 1 = (- 12 x + 5) + 12 x$

Simplify the arithmetic:

$13 x + 1 = (- 12 x + 5) + 12 x$

Group like terms:

$13 x + 1 = (- 12 x + 12 x) + 5$

Simplify the arithmetic:

$13 x + 1 = 5$

Subtract from both sides:

$(13 x + 1) - 1 = 5 - 1$

Simplify the arithmetic:

$13 x = 5 - 1$

Simplify the arithmetic:

$13 x = 4$

Divide both sides by :

$\frac{(13 x)}{13} = \frac{4}{13}$

Simplify the fraction:

$x = \frac{4}{13}$

3. List the solutions

$x = \frac{6}{11}, \frac{4}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x + 1 |$
$y = | 12 x - 5 |$
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