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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = - 0.333, 3

x=-0.333 , 3

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
$5 | - x + 1 | = | x + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| - x + 1 \| = \| x + 7 \|$
$x = + y$	$5 (- x + 1) = (x + 7)$
$x = - y$	$5 (- x + 1) = - (x + 7)$
$+ x = y$	$5 (- x + 1) = (x + 7)$
$- x = y$	$5 (- (- x + 1)) = (x + 7)$

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:

$\| x \| = \| y \|$	$5 \| - x + 1 \| = \| x + 7 \|$
$x = + y$ , $+ x = y$	$5 (- x + 1) = (x + 7)$
$x = - y$ , $- x = y$	$5 (- x + 1) = - (x + 7)$

2. Solve the two equations for x

17 additional steps

$5 \cdot (- x + 1) = (x + 7)$

Expand the parentheses:

$5 \cdot - x + 5 \cdot 1 = (x + 7)$

Group like terms:

$(5 \cdot - 1) x + 5 \cdot 1 = (x + 7)$

Multiply the coefficients:

$- 5 x + 5 \cdot 1 = (x + 7)$

Simplify the arithmetic:

$- 5 x + 5 = (x + 7)$

Subtract from both sides:

$(- 5 x + 5) - x = (x + 7) - x$

Group like terms:

$(- 5 x - x) + 5 = (x + 7) - x$

Simplify the arithmetic:

$- 6 x + 5 = (x + 7) - x$

Group like terms:

$- 6 x + 5 = (x - x) + 7$

Simplify the arithmetic:

$- 6 x + 5 = 7$

Subtract from both sides:

$(- 6 x + 5) - 5 = 7 - 5$

Simplify the arithmetic:

$- 6 x = 7 - 5$

Simplify the arithmetic:

$- 6 x = 2$

Divide both sides by :

$\frac{(- 6 x)}{- 6} = \frac{2}{- 6}$

Cancel out the negatives:

$\frac{6 x}{6} = \frac{2}{- 6}$

Simplify the fraction:

$x = \frac{2}{- 6}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 2}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{3}$

18 additional steps

$5 \cdot (- x + 1) = - (x + 7)$

Expand the parentheses:

$5 \cdot - x + 5 \cdot 1 = - (x + 7)$

Group like terms:

$(5 \cdot - 1) x + 5 \cdot 1 = - (x + 7)$

Multiply the coefficients:

$- 5 x + 5 \cdot 1 = - (x + 7)$

Simplify the arithmetic:

$- 5 x + 5 = - (x + 7)$

Expand the parentheses:

$- 5 x + 5 = - x - 7$

Add to both sides:

$(- 5 x + 5) + x = (- x - 7) + x$

Group like terms:

$(- 5 x + x) + 5 = (- x - 7) + x$

Simplify the arithmetic:

$- 4 x + 5 = (- x - 7) + x$

Group like terms:

$- 4 x + 5 = (- x + x) - 7$

Simplify the arithmetic:

$- 4 x + 5 = - 7$

Subtract from both sides:

$(- 4 x + 5) - 5 = - 7 - 5$

Simplify the arithmetic:

$- 4 x = - 7 - 5$

Simplify the arithmetic:

$- 4 x = - 12$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 12}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 12}{- 4}$

Simplify the fraction:

$x = \frac{- 12}{- 4}$

Cancel out the negatives:

$x = \frac{12}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = 3$

3. List the solutions

$x = - \frac{1}{3}, 3$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 5 | - x + 1 |$
$y = | x + 7 |$
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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