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Solution - Absolute value equations

Exact form:

x = - \frac{6}{7}, - \frac{14}{3}

x=-\frac{6}{7} , -\frac{14}{3}

Mixed number form:

x = - \frac{6}{7}, - 4 \frac{2}{3}

x=-\frac{6}{7} , -4\frac{2}{3}

Decimal form:

x = - 0.857, - 4.667

x=-0.857 , -4.667

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
$| - x + 2 | = 5 | \frac{1}{2} x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - x + 2 \| = 5 \| \frac{1}{2} x + 1 \|$
$x = + y$	$(- x + 2) = 5 (\frac{1}{2} x + 1)$
$x = - y$	$(- x + 2) = 5 (- (\frac{1}{2} x + 1))$
$+ x = y$	$(- x + 2) = 5 (\frac{1}{2} x + 1)$
$- x = y$	$- (- x + 2) = 5 (\frac{1}{2} x + 1)$

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:

$\| x \| = \| y \|$	$\| - x + 2 \| = 5 \| \frac{1}{2} x + 1 \|$
$x = + y$ , $+ x = y$	$(- x + 2) = 5 (\frac{1}{2} x + 1)$
$x = - y$ , $- x = y$	$(- x + 2) = 5 (- (\frac{1}{2} x + 1))$

2. Solve the two equations for x

26 additional steps

$(- x + 2) = 5 \cdot (\frac{1}{2} x + 1)$

Expand the parentheses:

$(- x + 2) = 5 \cdot \frac{1}{2} x + 5 \cdot 1$

Multiply the coefficients:

$(- x + 2) = \frac{(5 \cdot 1)}{2} x + 5 \cdot 1$

Simplify the arithmetic:

$(- x + 2) = \frac{(5 \cdot 1)}{2} x + 5$

Combine like terms:

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

Subtract from both sides:

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

Group like terms:

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

Group the coefficients:

$(- 1 + \frac{- 5}{2}) x + 2 = (\frac{5}{2} \cdot x + 5) - \frac{5}{2} x$

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

$\frac{- 7}{2} \cdot x + 2 = (\frac{5}{2} \cdot x + 5) - \frac{5}{2} x$

Group like terms:

$\frac{- 7}{2} \cdot x + 2 = (\frac{5}{2} \cdot x + \frac{- 5}{2} x) + 5$

Combine the fractions:

$\frac{- 7}{2} \cdot x + 2 = \frac{(5 - 5)}{2} x + 5$

Combine the numerators:

$\frac{- 7}{2} \cdot x + 2 = \frac{0}{2} x + 5$

Reduce the zero numerator:

$\frac{- 7}{2} x + 2 = 0 x + 5$

Simplify the arithmetic:

$\frac{- 7}{2} x + 2 = 5$

Subtract from both sides:

$(\frac{- 7}{2} x + 2) - 2 = 5 - 2$

Simplify the arithmetic:

$\frac{- 7}{2} x = 5 - 2$

Simplify the arithmetic:

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

Multiply both sides by inverse fraction :

$(\frac{- 7}{2} x) \cdot \frac{2}{- 7} = 3 \cdot \frac{2}{- 7}$

Move the negative sign from the denominator to the numerator:

$\frac{- 7}{2} x \cdot \frac{- 2}{7} = 3 \cdot \frac{2}{- 7}$

Group like terms:

$(\frac{- 7}{2} \cdot \frac{- 2}{7}) x = 3 \cdot \frac{2}{- 7}$

Multiply the coefficients:

$\frac{(- 7 \cdot - 2)}{(2 \cdot 7)} x = 3 \cdot \frac{2}{- 7}$

Simplify the arithmetic:

$1 x = 3 \cdot \frac{2}{- 7}$

$x = 3 \cdot \frac{2}{- 7}$

Move the negative sign from the denominator to the numerator:

$x = 3 \cdot \frac{- 2}{7}$

Multiply the fraction(s):

$x = \frac{(3 \cdot - 2)}{7}$

Simplify the arithmetic:

$x = \frac{- 6}{7}$

24 additional steps

$(- x + 2) = 5 \cdot (- (\frac{1}{2} x + 1))$

Expand the parentheses:

$(- x + 2) = 5 \cdot (\frac{- 1}{2} x - 1)$

Expand the parentheses:

$(- x + 2) = 5 \cdot \frac{- 1}{2} x + 5 \cdot - 1$

Multiply the coefficients:

$(- x + 2) = \frac{(5 \cdot - 1)}{2} x + 5 \cdot - 1$

Simplify the arithmetic:

$(- x + 2) = \frac{(5 \cdot - 1)}{2} x - 5$

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

Add to both sides:

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

Group like terms:

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

Group the coefficients:

$(- 1 + \frac{5}{2}) x + 2 = (\frac{- 5}{2} \cdot x - 5) + \frac{5}{2} x$

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

$\frac{3}{2} \cdot x + 2 = (\frac{- 5}{2} \cdot x - 5) + \frac{5}{2} x$

Group like terms:

$\frac{3}{2} \cdot x + 2 = (\frac{- 5}{2} \cdot x + \frac{5}{2} x) - 5$

Combine the fractions:

$\frac{3}{2} \cdot x + 2 = \frac{(- 5 + 5)}{2} x - 5$

Combine the numerators:

$\frac{3}{2} \cdot x + 2 = \frac{0}{2} x - 5$

Reduce the zero numerator:

$\frac{3}{2} x + 2 = 0 x - 5$

Simplify the arithmetic:

$\frac{3}{2} x + 2 = - 5$

Subtract from both sides:

$(\frac{3}{2} x + 2) - 2 = - 5 - 2$

Simplify the arithmetic:

$\frac{3}{2} x = - 5 - 2$

Simplify the arithmetic:

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

Multiply both sides by inverse fraction :

$(\frac{3}{2} x) \cdot \frac{2}{3} = - 7 \cdot \frac{2}{3}$

Group like terms:

$(\frac{3}{2} \cdot \frac{2}{3}) x = - 7 \cdot \frac{2}{3}$

Multiply the coefficients:

$\frac{(3 \cdot 2)}{(2 \cdot 3)} x = - 7 \cdot \frac{2}{3}$

Simplify the fraction:

$x = - 7 \cdot \frac{2}{3}$

Multiply the fraction(s):

$x = \frac{(- 7 \cdot 2)}{3}$

Simplify the arithmetic:

$x = \frac{- 14}{3}$

3. List the solutions

$x = - \frac{6}{7}, - \frac{14}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - x + 2 |$
$y = 5 | \frac{1}{2} x + 1 |$
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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