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Solution - Absolute value equations

Exact form:

x = 4, 8

x=4 , 8

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

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

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:

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

2. Solve the two equations for x

15 additional steps

$(- 2 x + 8) = 2 \cdot (x - 4)$

Expand the parentheses:

$(- 2 x + 8) = 2 x + 2 \cdot - 4$

Simplify the arithmetic:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 4 x + 8 = - 8$

Subtract from both sides:

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

Simplify the arithmetic:

$- 4 x = - 8 - 8$

Simplify the arithmetic:

$- 4 x = - 16$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 16}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 16}{- 4}$

Simplify the fraction:

$x = \frac{- 16}{- 4}$

Cancel out the negatives:

$x = \frac{16}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = 4$

9 additional steps

$(- 2 x + 8) = 2 \cdot (- (x - 4))$

Expand the parentheses:

$(- 2 x + 8) = 2 \cdot (- x + 4)$

$(- 2 x + 8) = 2 \cdot - x + 2 \cdot 4$

Group like terms:

$(- 2 x + 8) = (2 \cdot - 1) x + 2 \cdot 4$

Multiply the coefficients:

$(- 2 x + 8) = - 2 x + 2 \cdot 4$

Simplify the arithmetic:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$8 = 8$

3. List the solutions

$x = 4, 8$
(2 solution(s))

4. Graph

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