Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{1}{2}, - \frac{3}{8}

x=\frac{1}{2} , -\frac{3}{8}

Decimal form:

x = 0.5, - 0.375

x=0.5 , -0.375

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
$| - 3 x - 2 | = | - 5 x - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 x - 2 \| = \| - 5 x - 1 \|$
$x = + y$	$(- 3 x - 2) = (- 5 x - 1)$
$x = - y$	$(- 3 x - 2) = - (- 5 x - 1)$
$+ x = y$	$(- 3 x - 2) = (- 5 x - 1)$
$- x = y$	$- (- 3 x - 2) = (- 5 x - 1)$

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:

$\| x \| = \| y \|$	$\| - 3 x - 2 \| = \| - 5 x - 1 \|$
$x = + y$ , $+ x = y$	$(- 3 x - 2) = (- 5 x - 1)$
$x = - y$ , $- x = y$	$(- 3 x - 2) = - (- 5 x - 1)$

2. Solve the two equations for x

9 additional steps

$(- 3 x - 2) = (- 5 x - 1)$

Add to both sides:

$(- 3 x - 2) + 5 x = (- 5 x - 1) + 5 x$

Group like terms:

$(- 3 x + 5 x) - 2 = (- 5 x - 1) + 5 x$

Simplify the arithmetic:

$2 x - 2 = (- 5 x - 1) + 5 x$

Group like terms:

$2 x - 2 = (- 5 x + 5 x) - 1$

Simplify the arithmetic:

$2 x - 2 = - 1$

Add to both sides:

$(2 x - 2) + 2 = - 1 + 2$

Simplify the arithmetic:

$2 x = - 1 + 2$

Simplify the arithmetic:

$2 x = 1$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{1}{2}$

Simplify the fraction:

$x = \frac{1}{2}$

12 additional steps

$(- 3 x - 2) = - (- 5 x - 1)$

Expand the parentheses:

$(- 3 x - 2) = 5 x + 1$

Subtract from both sides:

$(- 3 x - 2) - 5 x = (5 x + 1) - 5 x$

Group like terms:

$(- 3 x - 5 x) - 2 = (5 x + 1) - 5 x$

Simplify the arithmetic:

$- 8 x - 2 = (5 x + 1) - 5 x$

Group like terms:

$- 8 x - 2 = (5 x - 5 x) + 1$

Simplify the arithmetic:

$- 8 x - 2 = 1$

Add to both sides:

$(- 8 x - 2) + 2 = 1 + 2$

Simplify the arithmetic:

$- 8 x = 1 + 2$

Simplify the arithmetic:

$- 8 x = 3$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

3. List the solutions

$x = \frac{1}{2}, - \frac{3}{8}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 3 x - 2 |$
$y = | - 5 x - 1 |$
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