Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = - \frac{3}{4}, \frac{1}{6}

x=-\frac{3}{4} , \frac{1}{6}

Decimal form:

x = - 0.75, 0.167

x=-0.75 , 0.167

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
$| 3 x - 6 | = 3 | 5 x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 x - 6 \| = 3 \| 5 x + 1 \|$
$x = + y$	$(3 x - 6) = 3 (5 x + 1)$
$x = - y$	$(3 x - 6) = 3 (- (5 x + 1))$
$+ x = y$	$(3 x - 6) = 3 (5 x + 1)$
$- x = y$	$- (3 x - 6) = 3 (5 x + 1)$

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:

$\| x \| = \| y \|$	$\| 3 x - 6 \| = 3 \| 5 x + 1 \|$
$x = + y$ , $+ x = y$	$(3 x - 6) = 3 (5 x + 1)$
$x = - y$ , $- x = y$	$(3 x - 6) = 3 (- (5 x + 1))$

2. Solve the two equations for x

16 additional steps

$(3 x - 6) = 3 \cdot (5 x + 1)$

Expand the parentheses:

$(3 x - 6) = 3 \cdot 5 x + 3 \cdot 1$

Multiply the coefficients:

$(3 x - 6) = 15 x + 3 \cdot 1$

Simplify the arithmetic:

$(3 x - 6) = 15 x + 3$

Subtract from both sides:

$(3 x - 6) - 15 x = (15 x + 3) - 15 x$

Group like terms:

$(3 x - 15 x) - 6 = (15 x + 3) - 15 x$

Simplify the arithmetic:

$- 12 x - 6 = (15 x + 3) - 15 x$

Group like terms:

$- 12 x - 6 = (15 x - 15 x) + 3$

Simplify the arithmetic:

$- 12 x - 6 = 3$

Add to both sides:

$(- 12 x - 6) + 6 = 3 + 6$

Simplify the arithmetic:

$- 12 x = 3 + 6$

Simplify the arithmetic:

$- 12 x = 9$

Divide both sides by :

$\frac{(- 12 x)}{- 12} = \frac{9}{- 12}$

Cancel out the negatives:

$\frac{12 x}{12} = \frac{9}{- 12}$

Simplify the fraction:

$x = \frac{9}{- 12}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 9}{12}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = \frac{- 3}{4}$

15 additional steps

$(3 x - 6) = 3 \cdot (- (5 x + 1))$

Expand the parentheses:

$(3 x - 6) = 3 \cdot (- 5 x - 1)$

Expand the parentheses:

$(3 x - 6) = 3 \cdot - 5 x + 3 \cdot - 1$

Multiply the coefficients:

$(3 x - 6) = - 15 x + 3 \cdot - 1$

Simplify the arithmetic:

$(3 x - 6) = - 15 x - 3$

Add to both sides:

$(3 x - 6) + 15 x = (- 15 x - 3) + 15 x$

Group like terms:

$(3 x + 15 x) - 6 = (- 15 x - 3) + 15 x$

Simplify the arithmetic:

$18 x - 6 = (- 15 x - 3) + 15 x$

Group like terms:

$18 x - 6 = (- 15 x + 15 x) - 3$

Simplify the arithmetic:

$18 x - 6 = - 3$

Add to both sides:

$(18 x - 6) + 6 = - 3 + 6$

Simplify the arithmetic:

$18 x = - 3 + 6$

Simplify the arithmetic:

$18 x = 3$

Divide both sides by :

$\frac{(18 x)}{18} = \frac{3}{18}$

Simplify the fraction:

$x = \frac{3}{18}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 3)}{(6 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{1}{6}$

3. List the solutions

$x = - \frac{3}{4}, \frac{1}{6}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 x - 6 |$
$y = 3 | 5 x + 1 |$
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