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Solution - Absolute value equations

Exact form:

w = \frac{1}{4}

w=\frac{1}{4}

Decimal form:

w = 0.25

w=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
$| 4 w | = | 4 w - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 w \| = \| 4 w - 2 \|$
$x = + y$	$(4 w) = (4 w - 2)$
$x = - y$	$(4 w) = - (4 w - 2)$
$+ x = y$	$(4 w) = (4 w - 2)$
$- x = y$	$- (4 w) = (4 w - 2)$

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:

$\| x \| = \| y \|$	$\| 4 w \| = \| 4 w - 2 \|$
$x = + y$ , $+ x = y$	$(4 w) = (4 w - 2)$
$x = - y$ , $- x = y$	$(4 w) = - (4 w - 2)$

2. Solve the two equations for w

4 additional steps

$4 w = (4 w - 2)$

Subtract from both sides:

$(4 w) - 4 w = (4 w - 2) - 4 w$

Simplify the arithmetic:

$0 = (4 w - 2) - 4 w$

Group like terms:

$0 = (4 w - 4 w) - 2$

Simplify the arithmetic:

$0 = - 2$

The statement is false:

$0 = - 2$

The equation is false so it has no solution.

8 additional steps

$4 w = - (4 w - 2)$

Expand the parentheses:

$4 w = - 4 w + 2$

Add to both sides:

$(4 w) + 4 w = (- 4 w + 2) + 4 w$

Simplify the arithmetic:

$8 w = (- 4 w + 2) + 4 w$

Group like terms:

$8 w = (- 4 w + 4 w) + 2$

Simplify the arithmetic:

$8 w = 2$

Divide both sides by :

$\frac{(8 w)}{8} = \frac{2}{8}$

Simplify the fraction:

$w = \frac{2}{8}$

Find the greatest common factor of the numerator and denominator:

$w = \frac{(1 \cdot 2)}{(4 \cdot 2)}$

Factor out and cancel the greatest common factor:

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

3. Graph

Each line represents the function of one side of the equation:
$y = | 4 w |$
$y = | 4 w - 2 |$
The equation is true where the two lines cross.

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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 w

3. 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 w

3. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved