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Solution - Absolute value equations

Exact form:

x = \frac{5}{2}

x=\frac{5}{2}

Mixed number form:

x = 2 \frac{1}{2}

x=2\frac{1}{2}

Decimal form:

x = 2.5

x=2.5

See steps

Step-by-step explanation

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

$| - x + 5 | - | x | = 0$

Add $| x |$ to both sides of the equation:

$| - x + 5 | - | x | + | x | = | x |$

Simplify the arithmetic

$| - x + 5 | = | x |$

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
$| - x + 5 | = | x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - x + 5 \| = \| x \|$
$x = + y$	$(- x + 5) = (x)$
$x = - y$	$(- x + 5) = (- (x))$
$+ x = y$	$(- x + 5) = (x)$
$- x = y$	$- (- x + 5) = (x)$

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:

$\| x \| = \| y \|$	$\| - x + 5 \| = \| x \|$
$x = + y$ , $+ x = y$	$(- x + 5) = (x)$
$x = - y$ , $- x = y$	$(- x + 5) = (- (x))$

3. Solve the two equations for x

10 additional steps

$(- x + 5) = x$

Subtract from both sides:

$(- x + 5) - x = x - x$

Group like terms:

$(- x - x) + 5 = x - x$

Simplify the arithmetic:

$- 2 x + 5 = x - x$

Simplify the arithmetic:

$- 2 x + 5 = 0$

Subtract from both sides:

$(- 2 x + 5) - 5 = 0 - 5$

Simplify the arithmetic:

$- 2 x = 0 - 5$

Simplify the arithmetic:

$- 2 x = - 5$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{- 5}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{- 5}{- 2}$

Simplify the fraction:

$x = \frac{- 5}{- 2}$

Cancel out the negatives:

$x = \frac{5}{2}$

4 additional steps

$(- x + 5) = - x$

Add to both sides:

$(- x + 5) + x = - x + x$

Group like terms:

$(- x + x) + 5 = - x + x$

Simplify the arithmetic:

$5 = - x + x$

Simplify the arithmetic:

$5 = 0$

The statement is false:

$5 = 0$

The equation is false so it has no solution.

4. List the solutions

$x = \frac{5}{2}$
(1 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - x + 5 |$
$y = | x |$
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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