Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = - 4, - 1

x=-4 , -1

See steps

Step-by-step explanation

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

$| - 3 x - 6 | - | - x + 2 | = 0$

Add $| - x + 2 |$ to both sides of the equation:

$| - 3 x - 6 | - | - x + 2 | + | - x + 2 | = | - x + 2 |$

Simplify the arithmetic

$| - 3 x - 6 | = | - x + 2 |$

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
$| - 3 x - 6 | = | - x + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 x - 6 \| = \| - x + 2 \|$
$x = + y$	$(- 3 x - 6) = (- x + 2)$
$x = - y$	$(- 3 x - 6) = (- (- x + 2))$
$+ x = y$	$(- 3 x - 6) = (- x + 2)$
$- x = y$	$- (- 3 x - 6) = (- x + 2)$

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:

$\| x \| = \| y \|$	$\| - 3 x - 6 \| = \| - x + 2 \|$
$x = + y$ , $+ x = y$	$(- 3 x - 6) = (- x + 2)$
$x = - y$ , $- x = y$	$(- 3 x - 6) = (- (- x + 2))$

3. Solve the two equations for x

13 additional steps

$(- 3 x - 6) = (- x + 2)$

Add to both sides:

$(- 3 x - 6) + x = (- x + 2) + x$

Group like terms:

$(- 3 x + x) - 6 = (- x + 2) + x$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 x - 6 = 2$

Add to both sides:

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

Simplify the arithmetic:

$- 2 x = 2 + 6$

Simplify the arithmetic:

$- 2 x = 8$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{8}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{8}{- 2}$

Simplify the fraction:

$x = \frac{8}{- 2}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 8}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 4 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = - 4$

13 additional steps

$(- 3 x - 6) = - (- x + 2)$

Expand the parentheses:

$(- 3 x - 6) = x - 2$

Subtract from both sides:

$(- 3 x - 6) - x = (x - 2) - x$

Group like terms:

$(- 3 x - x) - 6 = (x - 2) - x$

Simplify the arithmetic:

$- 4 x - 6 = (x - 2) - x$

Group like terms:

$- 4 x - 6 = (x - x) - 2$

Simplify the arithmetic:

$- 4 x - 6 = - 2$

Add to both sides:

$(- 4 x - 6) + 6 = - 2 + 6$

Simplify the arithmetic:

$- 4 x = - 2 + 6$

Simplify the arithmetic:

$- 4 x = 4$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

Simplify the fraction:

$x = - 1$

4. List the solutions

$x = - 4, - 1$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - 3 x - 6 |$
$y = | - x + 2 |$
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