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Solution - Absolute value equations

Exact form:

x = \frac{12}{625}, - \frac{12}{595}

x=\frac{12}{625} , -\frac{12}{595}

Decimal form:

x = 0.019, - 0.020

x=0.019 , -0.020

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
$| - 15 x + 12 | = | 610 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 15 x + 12 \| = \| 610 x \|$
$x = + y$	$(- 15 x + 12) = (610 x)$
$x = - y$	$(- 15 x + 12) = - (610 x)$
$+ x = y$	$(- 15 x + 12) = (610 x)$
$- x = y$	$- (- 15 x + 12) = (610 x)$

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:

$\| x \| = \| y \|$	$\| - 15 x + 12 \| = \| 610 x \|$
$x = + y$ , $+ x = y$	$(- 15 x + 12) = (610 x)$
$x = - y$ , $- x = y$	$(- 15 x + 12) = - (610 x)$

2. Solve the two equations for x

10 additional steps

$(- 15 x + 12) = 610 x$

Subtract from both sides:

$(- 15 x + 12) - 610 x = (610 x) - 610 x$

Group like terms:

$(- 15 x - 610 x) + 12 = (610 x) - 610 x$

Simplify the arithmetic:

$- 625 x + 12 = (610 x) - 610 x$

Simplify the arithmetic:

$- 625 x + 12 = 0$

Subtract from both sides:

$(- 625 x + 12) - 12 = 0 - 12$

Simplify the arithmetic:

$- 625 x = 0 - 12$

Simplify the arithmetic:

$- 625 x = - 12$

Divide both sides by :

$\frac{(- 625 x)}{- 625} = \frac{- 12}{- 625}$

Cancel out the negatives:

$\frac{625 x}{625} = \frac{- 12}{- 625}$

Simplify the fraction:

$x = \frac{- 12}{- 625}$

Cancel out the negatives:

$x = \frac{12}{625}$

7 additional steps

$(- 15 x + 12) = - 610 x$

Subtract from both sides:

$(- 15 x + 12) - 12 = (- 610 x) - 12$

Simplify the arithmetic:

$- 15 x = (- 610 x) - 12$

Add to both sides:

$(- 15 x) + 610 x = ((- 610 x) - 12) + 610 x$

Simplify the arithmetic:

$595 x = ((- 610 x) - 12) + 610 x$

Group like terms:

$595 x = (- 610 x + 610 x) - 12$

Simplify the arithmetic:

$595 x = - 12$

Divide both sides by :

$\frac{(595 x)}{595} = \frac{- 12}{595}$

Simplify the fraction:

$x = \frac{- 12}{595}$

3. List the solutions

$x = \frac{12}{625}, - \frac{12}{595}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 15 x + 12 |$
$y = | 610 x |$
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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