Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = 8, 4

x=8 , 4

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
$| - 2 x + 10 | = | - x + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 x + 10 \| = \| - x + 2 \|$
$x = + y$	$(- 2 x + 10) = (- x + 2)$
$x = - y$	$(- 2 x + 10) = - (- x + 2)$
$+ x = y$	$(- 2 x + 10) = (- x + 2)$
$- x = y$	$- (- 2 x + 10) = (- x + 2)$

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:

$\| x \| = \| y \|$	$\| - 2 x + 10 \| = \| - x + 2 \|$
$x = + y$ , $+ x = y$	$(- 2 x + 10) = (- x + 2)$
$x = - y$ , $- x = y$	$(- 2 x + 10) = - (- x + 2)$

2. Solve the two equations for x

10 additional steps

$(- 2 x + 10) = (- x + 2)$

Add to both sides:

$(- 2 x + 10) + x = (- x + 2) + x$

Group like terms:

$(- 2 x + x) + 10 = (- x + 2) + x$

Simplify the arithmetic:

$- x + 10 = (- x + 2) + x$

Group like terms:

$- x + 10 = (- x + x) + 2$

Simplify the arithmetic:

$- x + 10 = 2$

Subtract from both sides:

$(- x + 10) - 10 = 2 - 10$

Simplify the arithmetic:

$- x = 2 - 10$

Simplify the arithmetic:

$- x = - 8$

Multiply both sides by :

$- x \cdot - 1 = - 8 \cdot - 1$

Remove the one(s):

$x = - 8 \cdot - 1$

Simplify the arithmetic:

$x = 8$

14 additional steps

$(- 2 x + 10) = - (- x + 2)$

Expand the parentheses:

$(- 2 x + 10) = x - 2$

Subtract from both sides:

$(- 2 x + 10) - x = (x - 2) - x$

Group like terms:

$(- 2 x - x) + 10 = (x - 2) - x$

Simplify the arithmetic:

$- 3 x + 10 = (x - 2) - x$

Group like terms:

$- 3 x + 10 = (x - x) - 2$

Simplify the arithmetic:

$- 3 x + 10 = - 2$

Subtract from both sides:

$(- 3 x + 10) - 10 = - 2 - 10$

Simplify the arithmetic:

$- 3 x = - 2 - 10$

Simplify the arithmetic:

$- 3 x = - 12$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{- 12}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{- 12}{- 3}$

Simplify the fraction:

$x = \frac{- 12}{- 3}$

Cancel out the negatives:

$x = \frac{12}{3}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(4 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = 4$

3. List the solutions

$x = 8, 4$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 2 x + 10 |$
$y = | - x + 2 |$
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 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