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Solution - Absolute value equations

Exact form:

x = 4, - 4

x=4 , -4

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

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

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:

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

2. Solve the two equations for x

11 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 x - 4 = 4$

Add to both sides:

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

Simplify the arithmetic:

$2 x = 4 + 4$

Simplify the arithmetic:

$2 x = 8$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{8}{2}$

Simplify the fraction:

$x = \frac{8}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(4 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 4$

5 additional steps

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

Expand the parentheses:

$(x - 4) = x - 4$

Subtract from both sides:

$(x - 4) - x = (x - 4) - x$

Group like terms:

$(x - x) - 4 = (x - 4) - x$

Simplify the arithmetic:

$- 4 = (x - 4) - x$

Group like terms:

$- 4 = (x - x) - 4$

Simplify the arithmetic:

$- 4 = - 4$

3. List the solutions

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

4. Graph

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