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Solution - Absolute value equations

Exact form:

x = - \frac{5}{12}, \frac{4}{15}

x=-\frac{5}{12} , \frac{4}{15}

Decimal form:

x = - 0.417, 0.267

x=-0.417 , 0.267

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
$| 9 x + \frac{1}{3} | = | x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 x + \frac{1}{3} \| = \| x - 3 \|$
$x = + y$	$(9 x + \frac{1}{3}) = (x - 3)$
$x = - y$	$(9 x + \frac{1}{3}) = - (x - 3)$
$+ x = y$	$(9 x + \frac{1}{3}) = (x - 3)$
$- x = y$	$- (9 x + \frac{1}{3}) = (x - 3)$

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:

$\| x \| = \| y \|$	$\| 9 x + \frac{1}{3} \| = \| x - 3 \|$
$x = + y$ , $+ x = y$	$(9 x + \frac{1}{3}) = (x - 3)$
$x = - y$ , $- x = y$	$(9 x + \frac{1}{3}) = - (x - 3)$

2. Solve the two equations for x

16 additional steps

$(9 x + \frac{1}{3}) = (x - 3)$

Subtract from both sides:

$(9 x + \frac{1}{3}) - x = (x - 3) - x$

Group like terms:

$(9 x - x) + \frac{1}{3} = (x - 3) - x$

Simplify the arithmetic:

$8 x + \frac{1}{3} = (x - 3) - x$

Group like terms:

$8 x + \frac{1}{3} = (x - x) - 3$

Simplify the arithmetic:

$8 x + \frac{1}{3} = - 3$

Subtract from both sides:

$(8 x + \frac{1}{3}) - \frac{1}{3} = - 3 - \frac{1}{3}$

Combine the fractions:

$8 x + \frac{(1 - 1)}{3} = - 3 - \frac{1}{3}$

Combine the numerators:

$8 x + \frac{0}{3} = - 3 - \frac{1}{3}$

Reduce the zero numerator:

$8 x + 0 = - 3 - \frac{1}{3}$

Simplify the arithmetic:

$8 x = - 3 - \frac{1}{3}$

Convert the integer into a fraction:

$8 x = \frac{- 9}{3} + \frac{- 1}{3}$

Combine the fractions:

$8 x = \frac{(- 9 - 1)}{3}$

Combine the numerators:

$8 x = \frac{- 10}{3}$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{(\frac{- 10}{3})}{8}$

Simplify the fraction:

$x = \frac{(\frac{- 10}{3})}{8}$

Simplify the arithmetic:

$x = \frac{- 10}{(3 \cdot 8)}$

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

17 additional steps

$(9 x + \frac{1}{3}) = - (x - 3)$

Expand the parentheses:

$(9 x + \frac{1}{3}) = - x + 3$

Add to both sides:

$(9 x + \frac{1}{3}) + x = (- x + 3) + x$

Group like terms:

$(9 x + x) + \frac{1}{3} = (- x + 3) + x$

Simplify the arithmetic:

$10 x + \frac{1}{3} = (- x + 3) + x$

Group like terms:

$10 x + \frac{1}{3} = (- x + x) + 3$

Simplify the arithmetic:

$10 x + \frac{1}{3} = 3$

Subtract from both sides:

$(10 x + \frac{1}{3}) - \frac{1}{3} = 3 - \frac{1}{3}$

Combine the fractions:

$10 x + \frac{(1 - 1)}{3} = 3 - \frac{1}{3}$

Combine the numerators:

$10 x + \frac{0}{3} = 3 - \frac{1}{3}$

Reduce the zero numerator:

$10 x + 0 = 3 - \frac{1}{3}$

Simplify the arithmetic:

$10 x = 3 - \frac{1}{3}$

Convert the integer into a fraction:

$10 x = \frac{9}{3} + \frac{- 1}{3}$

Combine the fractions:

$10 x = \frac{(9 - 1)}{3}$

Combine the numerators:

$10 x = \frac{8}{3}$

Divide both sides by :

$\frac{(10 x)}{10} = \frac{(\frac{8}{3})}{10}$

Simplify the fraction:

$x = \frac{(\frac{8}{3})}{10}$

Simplify the arithmetic:

$x = \frac{8}{(3 \cdot 10)}$

$x = \frac{4}{15}$

3. List the solutions

$x = - \frac{5}{12}, \frac{4}{15}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 x + \frac{1}{3} |$
$y = | x - 3 |$
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 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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