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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = - 2, 0.4

x=-2 , 0.4

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
$| - x - 8 | = 2 | 3 x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - x - 8 \| = 2 \| 3 x + 3 \|$
$x = + y$	$(- x - 8) = 2 (3 x + 3)$
$x = - y$	$(- x - 8) = 2 (- (3 x + 3))$
$+ x = y$	$(- x - 8) = 2 (3 x + 3)$
$- x = y$	$- (- x - 8) = 2 (3 x + 3)$

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:

$\| x \| = \| y \|$	$\| - x - 8 \| = 2 \| 3 x + 3 \|$
$x = + y$ , $+ x = y$	$(- x - 8) = 2 (3 x + 3)$
$x = - y$ , $- x = y$	$(- x - 8) = 2 (- (3 x + 3))$

2. Solve the two equations for x

16 additional steps

$(- x - 8) = 2 \cdot (3 x + 3)$

Expand the parentheses:

$(- x - 8) = 2 \cdot 3 x + 2 \cdot 3$

Multiply the coefficients:

$(- x - 8) = 6 x + 2 \cdot 3$

Simplify the arithmetic:

$(- x - 8) = 6 x + 6$

Subtract from both sides:

$(- x - 8) - 6 x = (6 x + 6) - 6 x$

Group like terms:

$(- x - 6 x) - 8 = (6 x + 6) - 6 x$

Simplify the arithmetic:

$- 7 x - 8 = (6 x + 6) - 6 x$

Group like terms:

$- 7 x - 8 = (6 x - 6 x) + 6$

Simplify the arithmetic:

$- 7 x - 8 = 6$

Add to both sides:

$(- 7 x - 8) + 8 = 6 + 8$

Simplify the arithmetic:

$- 7 x = 6 + 8$

Simplify the arithmetic:

$- 7 x = 14$

Divide both sides by :

$\frac{(- 7 x)}{- 7} = \frac{14}{- 7}$

Cancel out the negatives:

$\frac{7 x}{7} = \frac{14}{- 7}$

Simplify the fraction:

$x = \frac{14}{- 7}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 14}{7}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 2 \cdot 7)}{(1 \cdot 7)}$

Factor out and cancel the greatest common factor:

$x = - 2$

13 additional steps

$(- x - 8) = 2 \cdot (- (3 x + 3))$

Expand the parentheses:

$(- x - 8) = 2 \cdot (- 3 x - 3)$

Expand the parentheses:

$(- x - 8) = 2 \cdot - 3 x + 2 \cdot - 3$

Multiply the coefficients:

$(- x - 8) = - 6 x + 2 \cdot - 3$

Simplify the arithmetic:

$(- x - 8) = - 6 x - 6$

Add to both sides:

$(- x - 8) + 6 x = (- 6 x - 6) + 6 x$

Group like terms:

$(- x + 6 x) - 8 = (- 6 x - 6) + 6 x$

Simplify the arithmetic:

$5 x - 8 = (- 6 x - 6) + 6 x$

Group like terms:

$5 x - 8 = (- 6 x + 6 x) - 6$

Simplify the arithmetic:

$5 x - 8 = - 6$

Add to both sides:

$(5 x - 8) + 8 = - 6 + 8$

Simplify the arithmetic:

$5 x = - 6 + 8$

Simplify the arithmetic:

$5 x = 2$

Divide both sides by :

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

Simplify the fraction:

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

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | - x - 8 |$
$y = 2 | 3 x + 3 |$
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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