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Solution - Absolute value equations

Exact form:

x = \frac{9}{2}

x=\frac{9}{2}

Mixed number form:

x = 4 \frac{1}{2}

x=4\frac{1}{2}

Decimal form:

x = 4.5

x=4.5

See steps

Step-by-step explanation

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

$| x - 4 | - | x - 5 | = 0$

Add $| x - 5 |$ to both sides of the equation:

$| x - 4 | - | x - 5 | + | x - 5 | = | x - 5 |$

Simplify the arithmetic

$| x - 4 | = | x - 5 |$

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
$| x - 4 | = | x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 4 \| = \| x - 5 \|$
$x = + y$	$(x - 4) = (x - 5)$
$x = - y$	$(x - 4) = (- (x - 5))$
$+ x = y$	$(x - 4) = (x - 5)$
$- x = y$	$- (x - 4) = (x - 5)$

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:

$\| x \| = \| y \|$	$\| x - 4 \| = \| x - 5 \|$
$x = + y$ , $+ x = y$	$(x - 4) = (x - 5)$
$x = - y$ , $- x = y$	$(x - 4) = (- (x - 5))$

3. Solve the two equations for x

5 additional steps

$(x - 4) = (x - 5)$

Subtract from both sides:

$(x - 4) - x = (x - 5) - x$

Group like terms:

$(x - x) - 4 = (x - 5) - x$

Simplify the arithmetic:

$- 4 = (x - 5) - x$

Group like terms:

$- 4 = (x - x) - 5$

Simplify the arithmetic:

$- 4 = - 5$

The statement is false:

$- 4 = - 5$

The equation is false so it has no solution.

10 additional steps

$(x - 4) = - (x - 5)$

Expand the parentheses:

$(x - 4) = - x + 5$

Add to both sides:

$(x - 4) + x = (- x + 5) + x$

Group like terms:

$(x + x) - 4 = (- x + 5) + x$

Simplify the arithmetic:

$2 x - 4 = (- x + 5) + x$

Group like terms:

$2 x - 4 = (- x + x) + 5$

Simplify the arithmetic:

$2 x - 4 = 5$

Add to both sides:

$(2 x - 4) + 4 = 5 + 4$

Simplify the arithmetic:

$2 x = 5 + 4$

Simplify the arithmetic:

$2 x = 9$

Divide both sides by :

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

Simplify the fraction:

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

4. Graph

Each line represents the function of one side of the equation:
$y = | x - 4 |$
$y = | x - 5 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

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