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Solution - Absolute value equations

Exact form:

x = \frac{1}{155}, \frac{1}{159}

x=\frac{1}{155} , \frac{1}{159}

Decimal form:

x = 0.006, 0.006

x=0.006 , 0.006

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
$| 4 x | = | 314 x - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x \| = \| 314 x - 2 \|$
$x = + y$	$(4 x) = (314 x - 2)$
$x = - y$	$(4 x) = - (314 x - 2)$
$+ x = y$	$(4 x) = (314 x - 2)$
$- x = y$	$- (4 x) = (314 x - 2)$

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:

$\| x \| = \| y \|$	$\| 4 x \| = \| 314 x - 2 \|$
$x = + y$ , $+ x = y$	$(4 x) = (314 x - 2)$
$x = - y$ , $- x = y$	$(4 x) = - (314 x - 2)$

2. Solve the two equations for x

9 additional steps

$4 x = (314 x - 2)$

Subtract from both sides:

$(4 x) - 314 x = (314 x - 2) - 314 x$

Simplify the arithmetic:

$- 310 x = (314 x - 2) - 314 x$

Group like terms:

$- 310 x = (314 x - 314 x) - 2$

Simplify the arithmetic:

$- 310 x = - 2$

Divide both sides by :

$\frac{(- 310 x)}{- 310} = \frac{- 2}{- 310}$

Cancel out the negatives:

$\frac{310 x}{310} = \frac{- 2}{- 310}$

Simplify the fraction:

$x = \frac{- 2}{- 310}$

Cancel out the negatives:

$x = \frac{2}{310}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 2)}{(155 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{1}{155}$

8 additional steps

$4 x = - (314 x - 2)$

Expand the parentheses:

$4 x = - 314 x + 2$

Add to both sides:

$(4 x) + 314 x = (- 314 x + 2) + 314 x$

Simplify the arithmetic:

$318 x = (- 314 x + 2) + 314 x$

Group like terms:

$318 x = (- 314 x + 314 x) + 2$

Simplify the arithmetic:

$318 x = 2$

Divide both sides by :

$\frac{(318 x)}{318} = \frac{2}{318}$

Simplify the fraction:

$x = \frac{2}{318}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 2)}{(159 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{1}{159}$

3. List the solutions

$x = \frac{1}{155}, \frac{1}{159}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x |$
$y = | 314 x - 2 |$
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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