Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

x = 2 \frac{1}{2}, 1

x=2\frac{1}{2} , 1

Decimal form:

x = 2.5, 1

x=2.5 , 1

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| - 2 x + 8 | + | - 6 x + 12 | = 0$

Add $- | - 6 x + 12 |$ to both sides of the equation:

$| - 2 x + 8 | + | - 6 x + 12 | - | - 6 x + 12 | = - | - 6 x + 12 |$

Simplify the arithmetic

$| - 2 x + 8 | = - | - 6 x + 12 |$

2. 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
$| - 2 x + 8 | = - | - 6 x + 12 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 x + 8 \| = - \| - 6 x + 12 \|$
$x = + y$	$(- 2 x + 8) = - (- 6 x + 12)$
$x = - y$	$(- 2 x + 8) = - - (- 6 x + 12)$
$+ x = y$	$(- 2 x + 8) = - (- 6 x + 12)$
$- x = y$	$- (- 2 x + 8) = - (- 6 x + 12)$

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:

$\| x \| = \| y \|$	$\| - 2 x + 8 \| = - \| - 6 x + 12 \|$
$x = + y$ , $+ x = y$	$(- 2 x + 8) = - (- 6 x + 12)$
$x = - y$ , $- x = y$	$(- 2 x + 8) = - - (- 6 x + 12)$

3. Solve the two equations for x

14 additional steps

$(- 2 x + 8) = - (- 6 x + 12)$

Expand the parentheses:

$(- 2 x + 8) = 6 x - 12$

Subtract from both sides:

$(- 2 x + 8) - 6 x = (6 x - 12) - 6 x$

Group like terms:

$(- 2 x - 6 x) + 8 = (6 x - 12) - 6 x$

Simplify the arithmetic:

$- 8 x + 8 = (6 x - 12) - 6 x$

Group like terms:

$- 8 x + 8 = (6 x - 6 x) - 12$

Simplify the arithmetic:

$- 8 x + 8 = - 12$

Subtract from both sides:

$(- 8 x + 8) - 8 = - 12 - 8$

Simplify the arithmetic:

$- 8 x = - 12 - 8$

Simplify the arithmetic:

$- 8 x = - 20$

Divide both sides by :

$\frac{(- 8 x)}{- 8} = \frac{- 20}{- 8}$

Cancel out the negatives:

$\frac{8 x}{8} = \frac{- 20}{- 8}$

Simplify the fraction:

$x = \frac{- 20}{- 8}$

Cancel out the negatives:

$x = \frac{20}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(5 \cdot 4)}{(2 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{5}{2}$

11 additional steps

$(- 2 x + 8) = - (- (- 6 x + 12))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(- 2 x + 8) = - 6 x + 12$

Add to both sides:

$(- 2 x + 8) + 6 x = (- 6 x + 12) + 6 x$

Group like terms:

$(- 2 x + 6 x) + 8 = (- 6 x + 12) + 6 x$

Simplify the arithmetic:

$4 x + 8 = (- 6 x + 12) + 6 x$

Group like terms:

$4 x + 8 = (- 6 x + 6 x) + 12$

Simplify the arithmetic:

$4 x + 8 = 12$

Subtract from both sides:

$(4 x + 8) - 8 = 12 - 8$

Simplify the arithmetic:

$4 x = 12 - 8$

Simplify the arithmetic:

$4 x = 4$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$x = 1$

4. List the solutions

$x = \frac{5}{2}, 1$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - 2 x + 8 |$
$y = - | - 6 x + 12 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved