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Solution - Absolute value equations

Exact form:

x = \frac{13}{4}, - \frac{1}{2}

x=\frac{13}{4} , -\frac{1}{2}

Mixed number form:

x = 3 \frac{1}{4}, - \frac{1}{2}

x=3\frac{1}{4} , -\frac{1}{2}

Decimal form:

x = 3.25, - 0.5

x=3.25 , -0.5

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
$| - 5 x + 5 | = | - x - 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 5 x + 5 \| = \| - x - 8 \|$
$x = + y$	$(- 5 x + 5) = (- x - 8)$
$x = - y$	$(- 5 x + 5) = - (- x - 8)$
$+ x = y$	$(- 5 x + 5) = (- x - 8)$
$- x = y$	$- (- 5 x + 5) = (- x - 8)$

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:

$\| x \| = \| y \|$	$\| - 5 x + 5 \| = \| - x - 8 \|$
$x = + y$ , $+ x = y$	$(- 5 x + 5) = (- x - 8)$
$x = - y$ , $- x = y$	$(- 5 x + 5) = - (- x - 8)$

2. Solve the two equations for x

11 additional steps

$(- 5 x + 5) = (- x - 8)$

Add to both sides:

$(- 5 x + 5) + x = (- x - 8) + x$

Group like terms:

$(- 5 x + x) + 5 = (- x - 8) + x$

Simplify the arithmetic:

$- 4 x + 5 = (- x - 8) + x$

Group like terms:

$- 4 x + 5 = (- x + x) - 8$

Simplify the arithmetic:

$- 4 x + 5 = - 8$

Subtract from both sides:

$(- 4 x + 5) - 5 = - 8 - 5$

Simplify the arithmetic:

$- 4 x = - 8 - 5$

Simplify the arithmetic:

$- 4 x = - 13$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

14 additional steps

$(- 5 x + 5) = - (- x - 8)$

Expand the parentheses:

$(- 5 x + 5) = x + 8$

Subtract from both sides:

$(- 5 x + 5) - x = (x + 8) - x$

Group like terms:

$(- 5 x - x) + 5 = (x + 8) - x$

Simplify the arithmetic:

$- 6 x + 5 = (x + 8) - x$

Group like terms:

$- 6 x + 5 = (x - x) + 8$

Simplify the arithmetic:

$- 6 x + 5 = 8$

Subtract from both sides:

$(- 6 x + 5) - 5 = 8 - 5$

Simplify the arithmetic:

$- 6 x = 8 - 5$

Simplify the arithmetic:

$- 6 x = 3$

Divide both sides by :

$\frac{(- 6 x)}{- 6} = \frac{3}{- 6}$

Cancel out the negatives:

$\frac{6 x}{6} = \frac{3}{- 6}$

Simplify the fraction:

$x = \frac{3}{- 6}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 3}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 3)}{(2 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{2}$

3. List the solutions

$x = \frac{13}{4}, - \frac{1}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 5 x + 5 |$
$y = | - x - 8 |$
The equation is true where the two lines cross.

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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

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