Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

w = - 8, - \frac{4}{5}

w=-8 , -\frac{4}{5}

Decimal form:

w = - 8, - 0.8

w=-8 , -0.8

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$2 | w - 1 | - 3 | w + 2 | = 0$

Add $3 | w + 2 |$ to both sides of the equation:

$2 | w - 1 | - 3 | w + 2 | + 3 | w + 2 | = 3 | w + 2 |$

Simplify the arithmetic

$2 | w - 1 | = 3 | w + 2 |$

2. 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
$2 | w - 1 | = 3 | w + 2 |$
without the absolute value bars:

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

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:

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

3. Solve the two equations for w

14 additional steps

$2 \cdot (w - 1) = 3 \cdot (w + 2)$

Expand the parentheses:

$2 w + 2 \cdot - 1 = 3 \cdot (w + 2)$

Simplify the arithmetic:

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

Expand the parentheses:

$2 w - 2 = 3 w + 3 \cdot 2$

Simplify the arithmetic:

$2 w - 2 = 3 w + 6$

Subtract from both sides:

$(2 w - 2) - 3 w = (3 w + 6) - 3 w$

Group like terms:

$(2 w - 3 w) - 2 = (3 w + 6) - 3 w$

Simplify the arithmetic:

$- w - 2 = (3 w + 6) - 3 w$

Group like terms:

$- w - 2 = (3 w - 3 w) + 6$

Simplify the arithmetic:

$- w - 2 = 6$

Add to both sides:

$(- w - 2) + 2 = 6 + 2$

Simplify the arithmetic:

$- w = 6 + 2$

Simplify the arithmetic:

$- w = 8$

Multiply both sides by :

$- w \cdot - 1 = 8 \cdot - 1$

Remove the one(s):

$w = 8 \cdot - 1$

Simplify the arithmetic:

$w = - 8$

16 additional steps

$2 \cdot (w - 1) = 3 \cdot (- (w + 2))$

Expand the parentheses:

$2 w + 2 \cdot - 1 = 3 \cdot (- (w + 2))$

Simplify the arithmetic:

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

Expand the parentheses:

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

$2 w - 2 = 3 \cdot - w + 3 \cdot - 2$

Group like terms:

$2 w - 2 = (3 \cdot - 1) w + 3 \cdot - 2$

Multiply the coefficients:

$2 w - 2 = - 3 w + 3 \cdot - 2$

Simplify the arithmetic:

$2 w - 2 = - 3 w - 6$

Add to both sides:

$(2 w - 2) + 3 w = (- 3 w - 6) + 3 w$

Group like terms:

$(2 w + 3 w) - 2 = (- 3 w - 6) + 3 w$

Simplify the arithmetic:

$5 w - 2 = (- 3 w - 6) + 3 w$

Group like terms:

$5 w - 2 = (- 3 w + 3 w) - 6$

Simplify the arithmetic:

$5 w - 2 = - 6$

Add to both sides:

$(5 w - 2) + 2 = - 6 + 2$

Simplify the arithmetic:

$5 w = - 6 + 2$

Simplify the arithmetic:

$5 w = - 4$

Divide both sides by :

$\frac{(5 w)}{5} = \frac{- 4}{5}$

Simplify the fraction:

$w = \frac{- 4}{5}$

4. List the solutions

$w = - 8, - \frac{4}{5}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 2 | w - 1 |$
$y = 3 | w + 2 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for w

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for w

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved