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Solution - Absolute value equations

Exact form:

x = \frac{9}{2}, - \frac{7}{6}

x=\frac{9}{2} , -\frac{7}{6}

Mixed number form:

x = 4 \frac{1}{2}, - 1 \frac{1}{6}

x=4\frac{1}{2} , -1\frac{1}{6}

Decimal form:

x = 4.5, - 1.167

x=4.5 , -1.167

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
$| 8 x - 2 | = | 4 x + 16 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 8 x - 2 \| = \| 4 x + 16 \|$
$x = + y$	$(8 x - 2) = (4 x + 16)$
$x = - y$	$(8 x - 2) = - (4 x + 16)$
$+ x = y$	$(8 x - 2) = (4 x + 16)$
$- x = y$	$- (8 x - 2) = (4 x + 16)$

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:

$\| x \| = \| y \|$	$\| 8 x - 2 \| = \| 4 x + 16 \|$
$x = + y$ , $+ x = y$	$(8 x - 2) = (4 x + 16)$
$x = - y$ , $- x = y$	$(8 x - 2) = - (4 x + 16)$

2. Solve the two equations for x

11 additional steps

$(8 x - 2) = (4 x + 16)$

Subtract from both sides:

$(8 x - 2) - 4 x = (4 x + 16) - 4 x$

Group like terms:

$(8 x - 4 x) - 2 = (4 x + 16) - 4 x$

Simplify the arithmetic:

$4 x - 2 = (4 x + 16) - 4 x$

Group like terms:

$4 x - 2 = (4 x - 4 x) + 16$

Simplify the arithmetic:

$4 x - 2 = 16$

Add to both sides:

$(4 x - 2) + 2 = 16 + 2$

Simplify the arithmetic:

$4 x = 16 + 2$

Simplify the arithmetic:

$4 x = 18$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{18}{4}$

Simplify the fraction:

$x = \frac{18}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(9 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{9}{2}$

12 additional steps

$(8 x - 2) = - (4 x + 16)$

Expand the parentheses:

$(8 x - 2) = - 4 x - 16$

Add to both sides:

$(8 x - 2) + 4 x = (- 4 x - 16) + 4 x$

Group like terms:

$(8 x + 4 x) - 2 = (- 4 x - 16) + 4 x$

Simplify the arithmetic:

$12 x - 2 = (- 4 x - 16) + 4 x$

Group like terms:

$12 x - 2 = (- 4 x + 4 x) - 16$

Simplify the arithmetic:

$12 x - 2 = - 16$

Add to both sides:

$(12 x - 2) + 2 = - 16 + 2$

Simplify the arithmetic:

$12 x = - 16 + 2$

Simplify the arithmetic:

$12 x = - 14$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{- 14}{12}$

Simplify the fraction:

$x = \frac{- 14}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 7 \cdot 2)}{(6 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 7}{6}$

3. List the solutions

$x = \frac{9}{2}, - \frac{7}{6}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 8 x - 2 |$
$y = | 4 x + 16 |$
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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