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Solution - Absolute value equations

Exact form:

x = \frac{5}{2}, \frac{5}{3}

x=\frac{5}{2} , \frac{5}{3}

Mixed number form:

x = 2 \frac{1}{2}, 1 \frac{2}{3}

x=2\frac{1}{2} , 1\frac{2}{3}

Decimal form:

x = 2.5, 1.667

x=2.5 , 1.667

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
$| x | = 5 | x - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x \| = 5 \| x - 2 \|$
$x = + y$	$(x) = 5 (x - 2)$
$x = - y$	$(x) = 5 (- (x - 2))$
$+ x = y$	$(x) = 5 (x - 2)$
$- x = y$	$- (x) = 5 (x - 2)$

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:

$\| x \| = \| y \|$	$\| x \| = 5 \| x - 2 \|$
$x = + y$ , $+ x = y$	$(x) = 5 (x - 2)$
$x = - y$ , $- x = y$	$(x) = 5 (- (x - 2))$

2. Solve the two equations for x

11 additional steps

$x = 5 \cdot (x - 2)$

Expand the parentheses:

$x = 5 x + 5 \cdot - 2$

Simplify the arithmetic:

$x = 5 x - 10$

Subtract from both sides:

$x - 5 x = (5 x - 10) - 5 x$

Simplify the arithmetic:

$- 4 x = (5 x - 10) - 5 x$

Group like terms:

$- 4 x = (5 x - 5 x) - 10$

Simplify the arithmetic:

$- 4 x = - 10$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 10}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 10}{- 4}$

Simplify the fraction:

$x = \frac{- 10}{- 4}$

Cancel out the negatives:

$x = \frac{10}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

12 additional steps

$x = 5 \cdot (- (x - 2))$

Expand the parentheses:

$x = 5 \cdot (- x + 2)$

$x = 5 \cdot - x + 5 \cdot 2$

Group like terms:

$x = (5 \cdot - 1) x + 5 \cdot 2$

Multiply the coefficients:

$x = - 5 x + 5 \cdot 2$

Simplify the arithmetic:

$x = - 5 x + 10$

Add to both sides:

$x + 5 x = (- 5 x + 10) + 5 x$

Simplify the arithmetic:

$6 x = (- 5 x + 10) + 5 x$

Group like terms:

$6 x = (- 5 x + 5 x) + 10$

Simplify the arithmetic:

$6 x = 10$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{10}{6}$

Simplify the fraction:

$x = \frac{10}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(5 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{5}{3}$

3. List the solutions

$x = \frac{5}{2}, \frac{5}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x |$
$y = 5 | x - 2 |$
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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