Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = - \frac{1}{3}, - \frac{1}{15}

x=-\frac{1}{3} , -\frac{1}{15}

Decimal form:

x = - 0.333, - 0.067

x=-0.333 , -0.067

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
$| 6 x | = | 9 x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 x \| = \| 9 x + 1 \|$
$x = + y$	$(6 x) = (9 x + 1)$
$x = - y$	$(6 x) = - (9 x + 1)$
$+ x = y$	$(6 x) = (9 x + 1)$
$- x = y$	$- (6 x) = (9 x + 1)$

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:

$\| x \| = \| y \|$	$\| 6 x \| = \| 9 x + 1 \|$
$x = + y$ , $+ x = y$	$(6 x) = (9 x + 1)$
$x = - y$ , $- x = y$	$(6 x) = - (9 x + 1)$

2. Solve the two equations for x

7 additional steps

$6 x = (9 x + 1)$

Subtract from both sides:

$(6 x) - 9 x = (9 x + 1) - 9 x$

Simplify the arithmetic:

$- 3 x = (9 x + 1) - 9 x$

Group like terms:

$- 3 x = (9 x - 9 x) + 1$

Simplify the arithmetic:

$- 3 x = 1$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

6 additional steps

$6 x = - (9 x + 1)$

Expand the parentheses:

$6 x = - 9 x - 1$

Add to both sides:

$(6 x) + 9 x = (- 9 x - 1) + 9 x$

Simplify the arithmetic:

$15 x = (- 9 x - 1) + 9 x$

Group like terms:

$15 x = (- 9 x + 9 x) - 1$

Simplify the arithmetic:

$15 x = - 1$

Divide both sides by :

$\frac{(15 x)}{15} = \frac{- 1}{15}$

Simplify the fraction:

$x = \frac{- 1}{15}$

3. List the solutions

$x = - \frac{1}{3}, - \frac{1}{15}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 x |$
$y = | 9 x + 1 |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

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

Latest Related Drills Solved