Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

= \frac{1}{4}, - \frac{1}{4}

=\frac{1}{4} , -\frac{1}{4}

Decimal form:

= 0.25, - 0.25

=0.25 , -0.25

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
$| + 3 | = 2 | 6 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| + 3 \| = 2 \| 6 x \|$
$x = + y$	$(+ 3) = 2 (6 x)$
$x = - y$	$(+ 3) = 2 (- (6 x))$
$+ x = y$	$(+ 3) = 2 (6 x)$
$- x = y$	$- (+ 3) = 2 (6 x)$

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:

$\| x \| = \| y \|$	$\| + 3 \| = 2 \| 6 x \|$
$x = + y$ , $+ x = y$	$(+ 3) = 2 (6 x)$
$x = - y$ , $- x = y$	$(+ 3) = 2 (- (6 x))$

2. Solve the two equations for

5 additional steps

$(3) = 2 \cdot 6 x$

Multiply the coefficients:

$(3) = 12 x$

Swap sides:

$12 x = (3)$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{(3)}{12}$

Simplify the fraction:

$x = \frac{(3)}{12}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = \frac{1}{4}$

7 additional steps

$(3) = 2 \cdot - 6 x$

Multiply the coefficients:

$(3) = - 12 x$

Swap sides:

$- 12 x = (3)$

Divide both sides by :

$\frac{(- 12 x)}{- 12} = \frac{(3)}{- 12}$

Cancel out the negatives:

$\frac{12 x}{12} = \frac{(3)}{- 12}$

Simplify the fraction:

$x = \frac{(3)}{- 12}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 3}{12}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{4}$

3. List the solutions

$= \frac{1}{4}, - \frac{1}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | + 3 |$
$y = 2 | 6 x |$
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

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

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved