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Solution - Absolute value equations

Exact form:

x = \frac{5}{12}, - \frac{1}{24}

x=\frac{5}{12} , -\frac{1}{24}

Decimal form:

x = 0.417, - 0.042

x=0.417 , -0.042

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

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

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:

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

2. Solve the two equations for x

18 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Add to both sides:

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

Combine the fractions:

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

Combine the numerators:

$2 x + \frac{0}{3} = (\frac{1}{2}) + \frac{1}{3}$

Reduce the zero numerator:

$2 x + 0 = (\frac{1}{2}) + \frac{1}{3}$

Simplify the arithmetic:

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

Find the lowest common denominator:

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

Multiply the denominators:

$2 x = \frac{(1 \cdot 3)}{6} + \frac{(1 \cdot 2)}{6}$

Multiply the numerators:

$2 x = \frac{3}{6} + \frac{2}{6}$

Combine the fractions:

$2 x = \frac{(3 + 2)}{6}$

Combine the numerators:

$2 x = \frac{5}{6}$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{(\frac{5}{6})}{2}$

Simplify the fraction:

$x = \frac{(\frac{5}{6})}{2}$

Simplify the arithmetic:

$x = \frac{5}{(6 \cdot 2)}$

$x = \frac{5}{12}$

19 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 x + \frac{- 1}{3} = \frac{- 1}{2}$

Add to both sides:

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

Combine the fractions:

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

Combine the numerators:

$4 x + \frac{0}{3} = (\frac{- 1}{2}) + \frac{1}{3}$

Reduce the zero numerator:

$4 x + 0 = (\frac{- 1}{2}) + \frac{1}{3}$

Simplify the arithmetic:

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

Find the lowest common denominator:

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

Multiply the denominators:

$4 x = \frac{(- 1 \cdot 3)}{6} + \frac{(1 \cdot 2)}{6}$

Multiply the numerators:

$4 x = \frac{- 3}{6} + \frac{2}{6}$

Combine the fractions:

$4 x = \frac{(- 3 + 2)}{6}$

Combine the numerators:

$4 x = \frac{- 1}{6}$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{(\frac{- 1}{6})}{4}$

Simplify the fraction:

$x = \frac{(\frac{- 1}{6})}{4}$

Simplify the arithmetic:

$x = \frac{- 1}{(6 \cdot 4)}$

$x = \frac{- 1}{24}$

3. List the solutions

$x = \frac{5}{12}, - \frac{1}{24}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 x - \frac{1}{3} |$
$y = | x + \frac{1}{2} |$
The equation is true where the two lines cross.

How did we do?

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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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