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Solution - Absolute value equations

Exact form:

x = - \frac{5}{18}, \frac{15}{2}

x=-\frac{5}{18} , \frac{15}{2}

Mixed number form:

x = - \frac{5}{18}, 7 \frac{1}{2}

x=-\frac{5}{18} , 7\frac{1}{2}

Decimal form:

x = - 0.278, 7.5

x=-0.278 , 7.5

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
$| 16 x + 20 | = | - 20 x + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 16 x + 20 \| = \| - 20 x + 10 \|$
$x = + y$	$(16 x + 20) = (- 20 x + 10)$
$x = - y$	$(16 x + 20) = - (- 20 x + 10)$
$+ x = y$	$(16 x + 20) = (- 20 x + 10)$
$- x = y$	$- (16 x + 20) = (- 20 x + 10)$

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:

$\| x \| = \| y \|$	$\| 16 x + 20 \| = \| - 20 x + 10 \|$
$x = + y$ , $+ x = y$	$(16 x + 20) = (- 20 x + 10)$
$x = - y$ , $- x = y$	$(16 x + 20) = - (- 20 x + 10)$

2. Solve the two equations for x

11 additional steps

$(16 x + 20) = (- 20 x + 10)$

Add to both sides:

$(16 x + 20) + 20 x = (- 20 x + 10) + 20 x$

Group like terms:

$(16 x + 20 x) + 20 = (- 20 x + 10) + 20 x$

Simplify the arithmetic:

$36 x + 20 = (- 20 x + 10) + 20 x$

Group like terms:

$36 x + 20 = (- 20 x + 20 x) + 10$

Simplify the arithmetic:

$36 x + 20 = 10$

Subtract from both sides:

$(36 x + 20) - 20 = 10 - 20$

Simplify the arithmetic:

$36 x = 10 - 20$

Simplify the arithmetic:

$36 x = - 10$

Divide both sides by :

$\frac{(36 x)}{36} = \frac{- 10}{36}$

Simplify the fraction:

$x = \frac{- 10}{36}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 2)}{(18 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 5}{18}$

14 additional steps

$(16 x + 20) = - (- 20 x + 10)$

Expand the parentheses:

$(16 x + 20) = 20 x - 10$

Subtract from both sides:

$(16 x + 20) - 20 x = (20 x - 10) - 20 x$

Group like terms:

$(16 x - 20 x) + 20 = (20 x - 10) - 20 x$

Simplify the arithmetic:

$- 4 x + 20 = (20 x - 10) - 20 x$

Group like terms:

$- 4 x + 20 = (20 x - 20 x) - 10$

Simplify the arithmetic:

$- 4 x + 20 = - 10$

Subtract from both sides:

$(- 4 x + 20) - 20 = - 10 - 20$

Simplify the arithmetic:

$- 4 x = - 10 - 20$

Simplify the arithmetic:

$- 4 x = - 30$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 30}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 30}{- 4}$

Simplify the fraction:

$x = \frac{- 30}{- 4}$

Cancel out the negatives:

$x = \frac{30}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(15 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{15}{2}$

3. List the solutions

$x = - \frac{5}{18}, \frac{15}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 16 x + 20 |$
$y = | - 20 x + 10 |$
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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