Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = 2, \frac{30}{11}

x=2 , \frac{30}{11}

Mixed number form:

x = 2, 2 \frac{8}{11}

x=2 , 2\frac{8}{11}

Decimal form:

x = 2, 2.727

x=2 , 2.727

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
$2 | 5 x - 12 | = | x - 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| 5 x - 12 \| = \| x - 6 \|$
$x = + y$	$2 (5 x - 12) = (x - 6)$
$x = - y$	$2 (5 x - 12) = - (x - 6)$
$+ x = y$	$2 (5 x - 12) = (x - 6)$
$- x = y$	$2 (- (5 x - 12)) = (x - 6)$

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:

$\| x \| = \| y \|$	$2 \| 5 x - 12 \| = \| x - 6 \|$
$x = + y$ , $+ x = y$	$2 (5 x - 12) = (x - 6)$
$x = - y$ , $- x = y$	$2 (5 x - 12) = - (x - 6)$

2. Solve the two equations for x

14 additional steps

$2 \cdot (5 x - 12) = (x - 6)$

Expand the parentheses:

$2 \cdot 5 x + 2 \cdot - 12 = (x - 6)$

Multiply the coefficients:

$10 x + 2 \cdot - 12 = (x - 6)$

Simplify the arithmetic:

$10 x - 24 = (x - 6)$

Subtract from both sides:

$(10 x - 24) - x = (x - 6) - x$

Group like terms:

$(10 x - x) - 24 = (x - 6) - x$

Simplify the arithmetic:

$9 x - 24 = (x - 6) - x$

Group like terms:

$9 x - 24 = (x - x) - 6$

Simplify the arithmetic:

$9 x - 24 = - 6$

Add to both sides:

$(9 x - 24) + 24 = - 6 + 24$

Simplify the arithmetic:

$9 x = - 6 + 24$

Simplify the arithmetic:

$9 x = 18$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{18}{9}$

Simplify the fraction:

$x = \frac{18}{9}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 9)}{(1 \cdot 9)}$

Factor out and cancel the greatest common factor:

$x = 2$

13 additional steps

$2 \cdot (5 x - 12) = - (x - 6)$

Expand the parentheses:

$2 \cdot 5 x + 2 \cdot - 12 = - (x - 6)$

Multiply the coefficients:

$10 x + 2 \cdot - 12 = - (x - 6)$

Simplify the arithmetic:

$10 x - 24 = - (x - 6)$

Expand the parentheses:

$10 x - 24 = - x + 6$

Add to both sides:

$(10 x - 24) + x = (- x + 6) + x$

Group like terms:

$(10 x + x) - 24 = (- x + 6) + x$

Simplify the arithmetic:

$11 x - 24 = (- x + 6) + x$

Group like terms:

$11 x - 24 = (- x + x) + 6$

Simplify the arithmetic:

$11 x - 24 = 6$

Add to both sides:

$(11 x - 24) + 24 = 6 + 24$

Simplify the arithmetic:

$11 x = 6 + 24$

Simplify the arithmetic:

$11 x = 30$

Divide both sides by :

$\frac{(11 x)}{11} = \frac{30}{11}$

Simplify the fraction:

$x = \frac{30}{11}$

3. List the solutions

$x = 2, \frac{30}{11}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 2 | 5 x - 12 |$
$y = | x - 6 |$
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