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Solution - Absolute value equations

Exact form:

x = 2

x=2

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| x | - | - x + 4 | = 0$

Add $| - x + 4 |$ to both sides of the equation:

$| x | - | - x + 4 | + | - x + 4 | = | - x + 4 |$

Simplify the arithmetic

$| x | = | - x + 4 |$

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

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

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:

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

3. Solve the two equations for x

7 additional steps

$x = (- x + 4)$

Add to both sides:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 x = 4$

Divide both sides by :

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

Simplify the fraction:

$x = \frac{4}{2}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = 2$

5 additional steps

$x = - (- x + 4)$

Expand the parentheses:

$x = x - 4$

Subtract from both sides:

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

Simplify the arithmetic:

$0 = (x - 4) - x$

Group like terms:

$0 = (x - x) - 4$

Simplify the arithmetic:

$0 = - 4$

The statement is false:

$0 = - 4$

The equation is false so it has no solution.

4. List the solutions

$x = 2$
(1 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | x |$
$y = | - x + 4 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

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