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Solution - Absolute value equations

Exact form:

w = - \frac{5}{18}

w=-\frac{5}{18}

Decimal form:

w = - 0.278

w=-0.278

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
$| w + \frac{7}{9} | = | w - \frac{2}{9} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| w + \frac{7}{9} \| = \| w - \frac{2}{9} \|$
$x = + y$	$(w + \frac{7}{9}) = (w - \frac{2}{9})$
$x = - y$	$(w + \frac{7}{9}) = - (w - \frac{2}{9})$
$+ x = y$	$(w + \frac{7}{9}) = (w - \frac{2}{9})$
$- x = y$	$- (w + \frac{7}{9}) = (w - \frac{2}{9})$

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:

$\| x \| = \| y \|$	$\| w + \frac{7}{9} \| = \| w - \frac{2}{9} \|$
$x = + y$ , $+ x = y$	$(w + \frac{7}{9}) = (w - \frac{2}{9})$
$x = - y$ , $- x = y$	$(w + \frac{7}{9}) = - (w - \frac{2}{9})$

2. Solve the two equations for w

5 additional steps

$(w + \frac{7}{9}) = (w + \frac{- 2}{9})$

Subtract from both sides:

$(w + \frac{7}{9}) - w = (w + \frac{- 2}{9}) - w$

Group like terms:

$(w - w) + \frac{7}{9} = (w + \frac{- 2}{9}) - w$

Simplify the arithmetic:

$\frac{7}{9} = (w + \frac{- 2}{9}) - w$

Group like terms:

$\frac{7}{9} = (w - w) + \frac{- 2}{9}$

Simplify the arithmetic:

$\frac{7}{9} = \frac{- 2}{9}$

The statement is false:

$\frac{7}{9} = \frac{- 2}{9}$

The equation is false so it has no solution.

16 additional steps

$(w + \frac{7}{9}) = - (w + \frac{- 2}{9})$

Expand the parentheses:

$(w + \frac{7}{9}) = - w + \frac{2}{9}$

Add to both sides:

$(w + \frac{7}{9}) + w = (- w + \frac{2}{9}) + w$

Group like terms:

$(w + w) + \frac{7}{9} = (- w + \frac{2}{9}) + w$

Simplify the arithmetic:

$2 w + \frac{7}{9} = (- w + \frac{2}{9}) + w$

Group like terms:

$2 w + \frac{7}{9} = (- w + w) + \frac{2}{9}$

Simplify the arithmetic:

$2 w + \frac{7}{9} = \frac{2}{9}$

Subtract from both sides:

$(2 w + \frac{7}{9}) - \frac{7}{9} = (\frac{2}{9}) - \frac{7}{9}$

Combine the fractions:

$2 w + \frac{(7 - 7)}{9} = (\frac{2}{9}) - \frac{7}{9}$

Combine the numerators:

$2 w + \frac{0}{9} = (\frac{2}{9}) - \frac{7}{9}$

Reduce the zero numerator:

$2 w + 0 = (\frac{2}{9}) - \frac{7}{9}$

Simplify the arithmetic:

$2 w = (\frac{2}{9}) - \frac{7}{9}$

Combine the fractions:

$2 w = \frac{(2 - 7)}{9}$

Combine the numerators:

$2 w = \frac{- 5}{9}$

Divide both sides by :

$\frac{(2 w)}{2} = \frac{(\frac{- 5}{9})}{2}$

Simplify the fraction:

$w = \frac{(\frac{- 5}{9})}{2}$

Simplify the arithmetic:

$w = \frac{- 5}{(9 \cdot 2)}$

$w = \frac{- 5}{18}$

3. Graph

Each line represents the function of one side of the equation:
$y = | w + \frac{7}{9} |$
$y = | w - \frac{2}{9} |$
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