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Solution - Absolute value equations

Exact form:

x = \frac{11}{12}

x=\frac{11}{12}

Mixed number form:

Decimal form:

x = 0.917

x=0.917

See steps

Step-by-step explanation

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

$| x + \frac{5}{6} | - | x - \frac{8}{3} | = 0$

Add $| x - \frac{8}{3} |$ to both sides of the equation:

$| x + \frac{5}{6} | - | x - \frac{8}{3} | + | x - \frac{8}{3} | = | x - \frac{8}{3} |$

Simplify the arithmetic

$| x + \frac{5}{6} | = | x - \frac{8}{3} |$

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
$| x + \frac{5}{6} | = | x - \frac{8}{3} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x + \frac{5}{6} \| = \| x - \frac{8}{3} \|$
$x = + y$	$(x + \frac{5}{6}) = (x - \frac{8}{3})$
$x = - y$	$(x + \frac{5}{6}) = (- (x - \frac{8}{3}))$
$+ x = y$	$(x + \frac{5}{6}) = (x - \frac{8}{3})$
$- x = y$	$- (x + \frac{5}{6}) = (x - \frac{8}{3})$

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:

$\| x \| = \| y \|$	$\| x + \frac{5}{6} \| = \| x - \frac{8}{3} \|$
$x = + y$ , $+ x = y$	$(x + \frac{5}{6}) = (x - \frac{8}{3})$
$x = - y$ , $- x = y$	$(x + \frac{5}{6}) = (- (x - \frac{8}{3}))$

3. Solve the two equations for x

5 additional steps

$(x + \frac{5}{6}) = (x + \frac{- 8}{3})$

Subtract from both sides:

$(x + \frac{5}{6}) - x = (x + \frac{- 8}{3}) - x$

Group like terms:

$(x - x) + \frac{5}{6} = (x + \frac{- 8}{3}) - x$

Simplify the arithmetic:

$\frac{5}{6} = (x + \frac{- 8}{3}) - x$

Group like terms:

$\frac{5}{6} = (x - x) + \frac{- 8}{3}$

Simplify the arithmetic:

$\frac{5}{6} = \frac{- 8}{3}$

The statement is false:

$\frac{5}{6} = \frac{- 8}{3}$

The equation is false so it has no solution.

19 additional steps

$(x + \frac{5}{6}) = - (x + \frac{- 8}{3})$

Expand the parentheses:

$(x + \frac{5}{6}) = - x + \frac{8}{3}$

Add to both sides:

$(x + \frac{5}{6}) + x = (- x + \frac{8}{3}) + x$

Group like terms:

$(x + x) + \frac{5}{6} = (- x + \frac{8}{3}) + x$

Simplify the arithmetic:

$2 x + \frac{5}{6} = (- x + \frac{8}{3}) + x$

Group like terms:

$2 x + \frac{5}{6} = (- x + x) + \frac{8}{3}$

Simplify the arithmetic:

$2 x + \frac{5}{6} = \frac{8}{3}$

Subtract from both sides:

$(2 x + \frac{5}{6}) - \frac{5}{6} = (\frac{8}{3}) - \frac{5}{6}$

Combine the fractions:

$2 x + \frac{(5 - 5)}{6} = (\frac{8}{3}) - \frac{5}{6}$

Combine the numerators:

$2 x + \frac{0}{6} = (\frac{8}{3}) - \frac{5}{6}$

Reduce the zero numerator:

$2 x + 0 = (\frac{8}{3}) - \frac{5}{6}$

Simplify the arithmetic:

$2 x = (\frac{8}{3}) - \frac{5}{6}$

Find the lowest common denominator:

$2 x = \frac{(8 \cdot 2)}{(3 \cdot 2)} + \frac{- 5}{6}$

Multiply the denominators:

$2 x = \frac{(8 \cdot 2)}{6} + \frac{- 5}{6}$

Multiply the numerators:

$2 x = \frac{16}{6} + \frac{- 5}{6}$

Combine the fractions:

$2 x = \frac{(16 - 5)}{6}$

Combine the numerators:

$2 x = \frac{11}{6}$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{(\frac{11}{6})}{2}$

Simplify the fraction:

$x = \frac{(\frac{11}{6})}{2}$

Simplify the arithmetic:

$x = \frac{11}{(6 \cdot 2)}$

$x = \frac{11}{12}$

4. Graph

Each line represents the function of one side of the equation:
$y = | x + \frac{5}{6} |$
$y = | x - \frac{8}{3} |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved