Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = - 0.6, - 3

x=-0.6 , -3

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

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

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:

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

2. Solve the two equations for x

12 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$(x - 3) = - 4 x - 2 \cdot 3$

Simplify the arithmetic:

$(x - 3) = - 4 x - 6$

Add to both sides:

$(x - 3) + 4 x = (- 4 x - 6) + 4 x$

Group like terms:

$(x + 4 x) - 3 = (- 4 x - 6) + 4 x$

Simplify the arithmetic:

$5 x - 3 = (- 4 x - 6) + 4 x$

Group like terms:

$5 x - 3 = (- 4 x + 4 x) - 6$

Simplify the arithmetic:

$5 x - 3 = - 6$

Add to both sides:

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

Simplify the arithmetic:

$5 x = - 6 + 3$

Simplify the arithmetic:

$5 x = - 3$

Divide both sides by :

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

Simplify the fraction:

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

17 additional steps

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

Expand the parentheses:

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

Expand the parentheses:

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

Multiply the coefficients:

$(x - 3) = 4 x - 2 \cdot - 3$

Simplify the arithmetic:

$(x - 3) = 4 x + 6$

Subtract from both sides:

$(x - 3) - 4 x = (4 x + 6) - 4 x$

Group like terms:

$(x - 4 x) - 3 = (4 x + 6) - 4 x$

Simplify the arithmetic:

$- 3 x - 3 = (4 x + 6) - 4 x$

Group like terms:

$- 3 x - 3 = (4 x - 4 x) + 6$

Simplify the arithmetic:

$- 3 x - 3 = 6$

Add to both sides:

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

Simplify the arithmetic:

$- 3 x = 6 + 3$

Simplify the arithmetic:

$- 3 x = 9$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{9}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{9}{- 3}$

Simplify the fraction:

$x = \frac{9}{- 3}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 9}{3}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = - 3$

3. List the solutions

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

4. Graph

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