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Solution - Absolute value equations

Exact form:

x = 9

x=9

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
$| x - 18 | = | x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 18 \| = \| x \|$
$x = + y$	$(x - 18) = (x)$
$x = - y$	$(x - 18) = - (x)$
$+ x = y$	$(x - 18) = (x)$
$- x = y$	$- (x - 18) = (x)$

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:

$\| x \| = \| y \|$	$\| x - 18 \| = \| x \|$
$x = + y$ , $+ x = y$	$(x - 18) = (x)$
$x = - y$ , $- x = y$	$(x - 18) = - (x)$

2. Solve the two equations for x

4 additional steps

$(x - 18) = x$

Subtract from both sides:

$(x - 18) - x = x - x$

Group like terms:

$(x - x) - 18 = x - x$

Simplify the arithmetic:

$- 18 = x - x$

Simplify the arithmetic:

$- 18 = 0$

The statement is false:

$- 18 = 0$

The equation is false so it has no solution.

10 additional steps

$(x - 18) = - x$

Add to both sides:

$(x - 18) + x = - x + x$

Group like terms:

$(x + x) - 18 = - x + x$

Simplify the arithmetic:

$2 x - 18 = - x + x$

Simplify the arithmetic:

$2 x - 18 = 0$

Add to both sides:

$(2 x - 18) + 18 = 0 + 18$

Simplify the arithmetic:

$2 x = 0 + 18$

Simplify the arithmetic:

$2 x = 18$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{18}{2}$

Simplify the fraction:

$x = \frac{18}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(9 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 9$

3. Graph

Each line represents the function of one side of the equation:
$y = | x - 18 |$
$y = | x |$
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 without absolute value bars

2. Solve the two equations for x

3. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved