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Solution - Absolute value equations

Exact form:

x = \frac{6}{25}, \frac{18}{5}

x=\frac{6}{25} , \frac{18}{5}

Mixed number form:

x = \frac{6}{25}, 3 \frac{3}{5}

x=\frac{6}{25} , 3\frac{3}{5}

Decimal form:

x = 0.24, 3.6

x=0.24 , 3.6

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
$| - 15 x + 12 | = | 10 x + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 15 x + 12 \| = \| 10 x + 6 \|$
$x = + y$	$(- 15 x + 12) = (10 x + 6)$
$x = - y$	$(- 15 x + 12) = - (10 x + 6)$
$+ x = y$	$(- 15 x + 12) = (10 x + 6)$
$- x = y$	$- (- 15 x + 12) = (10 x + 6)$

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:

$\| x \| = \| y \|$	$\| - 15 x + 12 \| = \| 10 x + 6 \|$
$x = + y$ , $+ x = y$	$(- 15 x + 12) = (10 x + 6)$
$x = - y$ , $- x = y$	$(- 15 x + 12) = - (10 x + 6)$

2. Solve the two equations for x

11 additional steps

$(- 15 x + 12) = (10 x + 6)$

Subtract from both sides:

$(- 15 x + 12) - 10 x = (10 x + 6) - 10 x$

Group like terms:

$(- 15 x - 10 x) + 12 = (10 x + 6) - 10 x$

Simplify the arithmetic:

$- 25 x + 12 = (10 x + 6) - 10 x$

Group like terms:

$- 25 x + 12 = (10 x - 10 x) + 6$

Simplify the arithmetic:

$- 25 x + 12 = 6$

Subtract from both sides:

$(- 25 x + 12) - 12 = 6 - 12$

Simplify the arithmetic:

$- 25 x = 6 - 12$

Simplify the arithmetic:

$- 25 x = - 6$

Divide both sides by :

$\frac{(- 25 x)}{- 25} = \frac{- 6}{- 25}$

Cancel out the negatives:

$\frac{25 x}{25} = \frac{- 6}{- 25}$

Simplify the fraction:

$x = \frac{- 6}{- 25}$

Cancel out the negatives:

$x = \frac{6}{25}$

12 additional steps

$(- 15 x + 12) = - (10 x + 6)$

Expand the parentheses:

$(- 15 x + 12) = - 10 x - 6$

Add to both sides:

$(- 15 x + 12) + 10 x = (- 10 x - 6) + 10 x$

Group like terms:

$(- 15 x + 10 x) + 12 = (- 10 x - 6) + 10 x$

Simplify the arithmetic:

$- 5 x + 12 = (- 10 x - 6) + 10 x$

Group like terms:

$- 5 x + 12 = (- 10 x + 10 x) - 6$

Simplify the arithmetic:

$- 5 x + 12 = - 6$

Subtract from both sides:

$(- 5 x + 12) - 12 = - 6 - 12$

Simplify the arithmetic:

$- 5 x = - 6 - 12$

Simplify the arithmetic:

$- 5 x = - 18$

Divide both sides by :

$\frac{(- 5 x)}{- 5} = \frac{- 18}{- 5}$

Cancel out the negatives:

$\frac{5 x}{5} = \frac{- 18}{- 5}$

Simplify the fraction:

$x = \frac{- 18}{- 5}$

Cancel out the negatives:

$x = \frac{18}{5}$

3. List the solutions

$x = \frac{6}{25}, \frac{18}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 15 x + 12 |$
$y = | 10 x + 6 |$
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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