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Solution - Absolute value equations

Exact form:

x = \frac{5}{24}, \frac{1}{36}

x=\frac{5}{24} , \frac{1}{36}

Decimal form:

x = 0.208, 0.028

x=0.208 , 0.028

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| 5 x - \frac{1}{2} | - | x + \frac{1}{3} | = 0$

Add $| x + \frac{1}{3} |$ to both sides of the equation:

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

Simplify the arithmetic

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

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

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

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:

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

3. Solve the two equations for x

18 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Add to both sides:

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

Combine the fractions:

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

Combine the numerators:

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

Reduce the zero numerator:

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

Simplify the arithmetic:

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

Find the lowest common denominator:

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

Multiply the denominators:

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

Multiply the numerators:

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

Combine the fractions:

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

Combine the numerators:

$4 x = \frac{5}{6}$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{(\frac{5}{6})}{4}$

Simplify the fraction:

$x = \frac{(\frac{5}{6})}{4}$

Simplify the arithmetic:

$x = \frac{5}{(6 \cdot 4)}$

$x = \frac{5}{24}$

19 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$6 x + \frac{- 1}{2} = \frac{- 1}{3}$

Add to both sides:

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

Combine the fractions:

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

Combine the numerators:

$6 x + \frac{0}{2} = (\frac{- 1}{3}) + \frac{1}{2}$

Reduce the zero numerator:

$6 x + 0 = (\frac{- 1}{3}) + \frac{1}{2}$

Simplify the arithmetic:

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

Find the lowest common denominator:

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

Multiply the denominators:

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

Multiply the numerators:

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

Combine the fractions:

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

Combine the numerators:

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

Divide both sides by :

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

Simplify the fraction:

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

Simplify the arithmetic:

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

$x = \frac{1}{36}$

4. List the solutions

$x = \frac{5}{24}, \frac{1}{36}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 5 x - \frac{1}{2} |$
$y = | x + \frac{1}{3} |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved