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Solution - Absolute value equations

Exact form:

x = \frac{9}{2}, - \frac{21}{4}

x=\frac{9}{2} , -\frac{21}{4}

Mixed number form:

x = 4 \frac{1}{2}, - 5 \frac{1}{4}

x=4\frac{1}{2} , -5\frac{1}{4}

Decimal form:

x = 4.5, - 5.25

x=4.5 , -5.25

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
$| x + 2 | = | \frac{1}{3} x + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x + 2 \| = \| \frac{1}{3} x + 5 \|$
$x = + y$	$(x + 2) = (\frac{1}{3} x + 5)$
$x = - y$	$(x + 2) = - (\frac{1}{3} x + 5)$
$+ x = y$	$(x + 2) = (\frac{1}{3} x + 5)$
$- x = y$	$- (x + 2) = (\frac{1}{3} x + 5)$

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:

$\| x \| = \| y \|$	$\| x + 2 \| = \| \frac{1}{3} x + 5 \|$
$x = + y$ , $+ x = y$	$(x + 2) = (\frac{1}{3} x + 5)$
$x = - y$ , $- x = y$	$(x + 2) = - (\frac{1}{3} x + 5)$

2. Solve the two equations for x

19 additional steps

$(x + 2) = (\frac{1}{3} x + 5)$

Subtract from both sides:

$(x + 2) - \frac{1}{3} \cdot x = (\frac{1}{3} x + 5) - \frac{1}{3} x$

Group like terms:

$(x + \frac{- 1}{3} \cdot x) + 2 = (\frac{1}{3} \cdot x + 5) - \frac{1}{3} x$

Group the coefficients:

$(1 + \frac{- 1}{3}) x + 2 = (\frac{1}{3} \cdot x + 5) - \frac{1}{3} x$

Convert the integer into a fraction:

$(\frac{3}{3} + \frac{- 1}{3}) x + 2 = (\frac{1}{3} \cdot x + 5) - \frac{1}{3} x$

Combine the fractions:

$\frac{(3 - 1)}{3} \cdot x + 2 = (\frac{1}{3} \cdot x + 5) - \frac{1}{3} x$

Combine the numerators:

$\frac{2}{3} \cdot x + 2 = (\frac{1}{3} \cdot x + 5) - \frac{1}{3} x$

Group like terms:

$\frac{2}{3} \cdot x + 2 = (\frac{1}{3} \cdot x + \frac{- 1}{3} x) + 5$

Combine the fractions:

$\frac{2}{3} \cdot x + 2 = \frac{(1 - 1)}{3} x + 5$

Combine the numerators:

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

Reduce the zero numerator:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

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

Multiply both sides by inverse fraction :

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the fraction:

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

Multiply the fraction(s):

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

Simplify the arithmetic:

$x = \frac{9}{2}$

20 additional steps

$(x + 2) = - (\frac{1}{3} x + 5)$

Expand the parentheses:

$(x + 2) = \frac{- 1}{3} x - 5$

Add to both sides:

$(x + 2) + \frac{1}{3} \cdot x = (\frac{- 1}{3} x - 5) + \frac{1}{3} x$

Group like terms:

$(x + \frac{1}{3} \cdot x) + 2 = (\frac{- 1}{3} \cdot x - 5) + \frac{1}{3} x$

Group the coefficients:

$(1 + \frac{1}{3}) x + 2 = (\frac{- 1}{3} \cdot x - 5) + \frac{1}{3} x$

Convert the integer into a fraction:

$(\frac{3}{3} + \frac{1}{3}) x + 2 = (\frac{- 1}{3} \cdot x - 5) + \frac{1}{3} x$

Combine the fractions:

$\frac{(3 + 1)}{3} \cdot x + 2 = (\frac{- 1}{3} \cdot x - 5) + \frac{1}{3} x$

Combine the numerators:

$\frac{4}{3} \cdot x + 2 = (\frac{- 1}{3} \cdot x - 5) + \frac{1}{3} x$

Group like terms:

$\frac{4}{3} \cdot x + 2 = (\frac{- 1}{3} \cdot x + \frac{1}{3} x) - 5$

Combine the fractions:

$\frac{4}{3} \cdot x + 2 = \frac{(- 1 + 1)}{3} x - 5$

Combine the numerators:

$\frac{4}{3} \cdot x + 2 = \frac{0}{3} x - 5$

Reduce the zero numerator:

$\frac{4}{3} x + 2 = 0 x - 5$

Simplify the arithmetic:

$\frac{4}{3} x + 2 = - 5$

Subtract from both sides:

$(\frac{4}{3} x + 2) - 2 = - 5 - 2$

Simplify the arithmetic:

$\frac{4}{3} x = - 5 - 2$

Simplify the arithmetic:

$\frac{4}{3} x = - 7$

Multiply both sides by inverse fraction :

$(\frac{4}{3} x) \cdot \frac{3}{4} = - 7 \cdot \frac{3}{4}$

Group like terms:

$(\frac{4}{3} \cdot \frac{3}{4}) x = - 7 \cdot \frac{3}{4}$

Multiply the coefficients:

$\frac{(4 \cdot 3)}{(3 \cdot 4)} x = - 7 \cdot \frac{3}{4}$

Simplify the fraction:

$x = - 7 \cdot \frac{3}{4}$

Multiply the fraction(s):

$x = \frac{(- 7 \cdot 3)}{4}$

Simplify the arithmetic:

$x = \frac{- 21}{4}$

3. List the solutions

$x = \frac{9}{2}, - \frac{21}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x + 2 |$
$y = | \frac{1}{3} x + 5 |$
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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