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Solution - Absolute value equations

Exact form:

x = - \frac{5}{3}, - \frac{1}{3}

x=-\frac{5}{3} , -\frac{1}{3}

Mixed number form:

x = - 1 \frac{2}{3}, - \frac{1}{3}

x=-1\frac{2}{3} , -\frac{1}{3}

Decimal form:

x = - 1.667, - 0.333

x=-1.667 , -0.333

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
$| 9 x + 5 | = | 6 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 x + 5 \| = \| 6 x \|$
$x = + y$	$(9 x + 5) = (6 x)$
$x = - y$	$(9 x + 5) = - (6 x)$
$+ x = y$	$(9 x + 5) = (6 x)$
$- x = y$	$- (9 x + 5) = (6 x)$

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:

$\| x \| = \| y \|$	$\| 9 x + 5 \| = \| 6 x \|$
$x = + y$ , $+ x = y$	$(9 x + 5) = (6 x)$
$x = - y$ , $- x = y$	$(9 x + 5) = - (6 x)$

2. Solve the two equations for x

8 additional steps

$(9 x + 5) = 6 x$

Subtract from both sides:

$(9 x + 5) - 6 x = (6 x) - 6 x$

Group like terms:

$(9 x - 6 x) + 5 = (6 x) - 6 x$

Simplify the arithmetic:

$3 x + 5 = (6 x) - 6 x$

Simplify the arithmetic:

$3 x + 5 = 0$

Subtract from both sides:

$(3 x + 5) - 5 = 0 - 5$

Simplify the arithmetic:

$3 x = 0 - 5$

Simplify the arithmetic:

$3 x = - 5$

Divide both sides by :

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

Simplify the fraction:

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

9 additional steps

$(9 x + 5) = - 6 x$

Subtract from both sides:

$(9 x + 5) - 5 = (- 6 x) - 5$

Simplify the arithmetic:

$9 x = (- 6 x) - 5$

Add to both sides:

$(9 x) + 6 x = ((- 6 x) - 5) + 6 x$

Simplify the arithmetic:

$15 x = ((- 6 x) - 5) + 6 x$

Group like terms:

$15 x = (- 6 x + 6 x) - 5$

Simplify the arithmetic:

$15 x = - 5$

Divide both sides by :

$\frac{(15 x)}{15} = \frac{- 5}{15}$

Simplify the fraction:

$x = \frac{- 5}{15}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

$x = - \frac{5}{3}, - \frac{1}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 x + 5 |$
$y = | 6 x |$
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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