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Solution - Absolute value equations

Exact form:

x = - \frac{1}{4}, - \frac{13}{8}

x=-\frac{1}{4} , -\frac{13}{8}

Mixed number form:

x = - \frac{1}{4}, - 1 \frac{5}{8}

x=-\frac{1}{4} , -1\frac{5}{8}

Decimal form:

x = - 0.25, - 1.625

x=-0.25 , -1.625

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
$| 2 x - 5 | = | - 10 x - 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 5 \| = \| - 10 x - 8 \|$
$x = + y$	$(2 x - 5) = (- 10 x - 8)$
$x = - y$	$(2 x - 5) = - (- 10 x - 8)$
$+ x = y$	$(2 x - 5) = (- 10 x - 8)$
$- x = y$	$- (2 x - 5) = (- 10 x - 8)$

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:

$\| x \| = \| y \|$	$\| 2 x - 5 \| = \| - 10 x - 8 \|$
$x = + y$ , $+ x = y$	$(2 x - 5) = (- 10 x - 8)$
$x = - y$ , $- x = y$	$(2 x - 5) = - (- 10 x - 8)$

2. Solve the two equations for x

11 additional steps

$(2 x - 5) = (- 10 x - 8)$

Add to both sides:

$(2 x - 5) + 10 x = (- 10 x - 8) + 10 x$

Group like terms:

$(2 x + 10 x) - 5 = (- 10 x - 8) + 10 x$

Simplify the arithmetic:

$12 x - 5 = (- 10 x - 8) + 10 x$

Group like terms:

$12 x - 5 = (- 10 x + 10 x) - 8$

Simplify the arithmetic:

$12 x - 5 = - 8$

Add to both sides:

$(12 x - 5) + 5 = - 8 + 5$

Simplify the arithmetic:

$12 x = - 8 + 5$

Simplify the arithmetic:

$12 x = - 3$

Divide both sides by :

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

Simplify the fraction:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{4}$

12 additional steps

$(2 x - 5) = - (- 10 x - 8)$

Expand the parentheses:

$(2 x - 5) = 10 x + 8$

Subtract from both sides:

$(2 x - 5) - 10 x = (10 x + 8) - 10 x$

Group like terms:

$(2 x - 10 x) - 5 = (10 x + 8) - 10 x$

Simplify the arithmetic:

$- 8 x - 5 = (10 x + 8) - 10 x$

Group like terms:

$- 8 x - 5 = (10 x - 10 x) + 8$

Simplify the arithmetic:

$- 8 x - 5 = 8$

Add to both sides:

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

Simplify the arithmetic:

$- 8 x = 8 + 5$

Simplify the arithmetic:

$- 8 x = 13$

Divide both sides by :

$\frac{(- 8 x)}{- 8} = \frac{13}{- 8}$

Cancel out the negatives:

$\frac{8 x}{8} = \frac{13}{- 8}$

Simplify the fraction:

$x = \frac{13}{- 8}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 13}{8}$

3. List the solutions

$x = - \frac{1}{4}, - \frac{13}{8}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 5 |$
$y = | - 10 x - 8 |$
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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