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Solution - Absolute value equations

Exact form:

x = - 6, \frac{29}{6}

x=-6 , \frac{29}{6}

Mixed number form:

x = - 6, 4 \frac{5}{6}

x=-6 , 4\frac{5}{6}

Decimal form:

x = - 6, 4.833

x=-6 , 4.833

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
$| - x - 71 | = | 13 x + 13 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - x - 71 \| = \| 13 x + 13 \|$
$x = + y$	$(- x - 71) = (13 x + 13)$
$x = - y$	$(- x - 71) = - (13 x + 13)$
$+ x = y$	$(- x - 71) = (13 x + 13)$
$- x = y$	$- (- x - 71) = (13 x + 13)$

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:

$\| x \| = \| y \|$	$\| - x - 71 \| = \| 13 x + 13 \|$
$x = + y$ , $+ x = y$	$(- x - 71) = (13 x + 13)$
$x = - y$ , $- x = y$	$(- x - 71) = - (13 x + 13)$

2. Solve the two equations for x

13 additional steps

$(- x - 71) = (13 x + 13)$

Subtract from both sides:

$(- x - 71) - 13 x = (13 x + 13) - 13 x$

Group like terms:

$(- x - 13 x) - 71 = (13 x + 13) - 13 x$

Simplify the arithmetic:

$- 14 x - 71 = (13 x + 13) - 13 x$

Group like terms:

$- 14 x - 71 = (13 x - 13 x) + 13$

Simplify the arithmetic:

$- 14 x - 71 = 13$

Add to both sides:

$(- 14 x - 71) + 71 = 13 + 71$

Simplify the arithmetic:

$- 14 x = 13 + 71$

Simplify the arithmetic:

$- 14 x = 84$

Divide both sides by :

$\frac{(- 14 x)}{- 14} = \frac{84}{- 14}$

Cancel out the negatives:

$\frac{14 x}{14} = \frac{84}{- 14}$

Simplify the fraction:

$x = \frac{84}{- 14}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 84}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 6 \cdot 14)}{(1 \cdot 14)}$

Factor out and cancel the greatest common factor:

$x = - 6$

12 additional steps

$(- x - 71) = - (13 x + 13)$

Expand the parentheses:

$(- x - 71) = - 13 x - 13$

Add to both sides:

$(- x - 71) + 13 x = (- 13 x - 13) + 13 x$

Group like terms:

$(- x + 13 x) - 71 = (- 13 x - 13) + 13 x$

Simplify the arithmetic:

$12 x - 71 = (- 13 x - 13) + 13 x$

Group like terms:

$12 x - 71 = (- 13 x + 13 x) - 13$

Simplify the arithmetic:

$12 x - 71 = - 13$

Add to both sides:

$(12 x - 71) + 71 = - 13 + 71$

Simplify the arithmetic:

$12 x = - 13 + 71$

Simplify the arithmetic:

$12 x = 58$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{58}{12}$

Simplify the fraction:

$x = \frac{58}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(29 \cdot 2)}{(6 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{29}{6}$

3. List the solutions

$x = - 6, \frac{29}{6}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - x - 71 |$
$y = | 13 x + 13 |$
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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