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Solution - Absolute value equations

Exact form:

x = - \frac{7}{2}, - \frac{2}{3}

x=-\frac{7}{2} , -\frac{2}{3}

Mixed number form:

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

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

Decimal form:

x = - 3.5, - 0.667

x=-3.5 , -0.667

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
$| 2 x - 10 | = | 10 x + 18 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 10 \| = \| 10 x + 18 \|$
$x = + y$	$(2 x - 10) = (10 x + 18)$
$x = - y$	$(2 x - 10) = - (10 x + 18)$
$+ x = y$	$(2 x - 10) = (10 x + 18)$
$- x = y$	$- (2 x - 10) = (10 x + 18)$

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:

$\| x \| = \| y \|$	$\| 2 x - 10 \| = \| 10 x + 18 \|$
$x = + y$ , $+ x = y$	$(2 x - 10) = (10 x + 18)$
$x = - y$ , $- x = y$	$(2 x - 10) = - (10 x + 18)$

2. Solve the two equations for x

13 additional steps

$(2 x - 10) = (10 x + 18)$

Subtract from both sides:

$(2 x - 10) - 10 x = (10 x + 18) - 10 x$

Group like terms:

$(2 x - 10 x) - 10 = (10 x + 18) - 10 x$

Simplify the arithmetic:

$- 8 x - 10 = (10 x + 18) - 10 x$

Group like terms:

$- 8 x - 10 = (10 x - 10 x) + 18$

Simplify the arithmetic:

$- 8 x - 10 = 18$

Add to both sides:

$(- 8 x - 10) + 10 = 18 + 10$

Simplify the arithmetic:

$- 8 x = 18 + 10$

Simplify the arithmetic:

$- 8 x = 28$

Divide both sides by :

$\frac{(- 8 x)}{- 8} = \frac{28}{- 8}$

Cancel out the negatives:

$\frac{8 x}{8} = \frac{28}{- 8}$

Simplify the fraction:

$x = \frac{28}{- 8}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 28}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 7 \cdot 4)}{(2 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 7}{2}$

12 additional steps

$(2 x - 10) = - (10 x + 18)$

Expand the parentheses:

$(2 x - 10) = - 10 x - 18$

Add to both sides:

$(2 x - 10) + 10 x = (- 10 x - 18) + 10 x$

Group like terms:

$(2 x + 10 x) - 10 = (- 10 x - 18) + 10 x$

Simplify the arithmetic:

$12 x - 10 = (- 10 x - 18) + 10 x$

Group like terms:

$12 x - 10 = (- 10 x + 10 x) - 18$

Simplify the arithmetic:

$12 x - 10 = - 18$

Add to both sides:

$(12 x - 10) + 10 = - 18 + 10$

Simplify the arithmetic:

$12 x = - 18 + 10$

Simplify the arithmetic:

$12 x = - 8$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{- 8}{12}$

Simplify the fraction:

$x = \frac{- 8}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 2 \cdot 4)}{(3 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 2}{3}$

3. List the solutions

$x = - \frac{7}{2}, - \frac{2}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 10 |$
$y = | 10 x + 18 |$
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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