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Solution - Absolute value equations

Exact form:

w = - \frac{7}{4}

w=-\frac{7}{4}

Mixed number form:

w = - 1 \frac{3}{4}

w=-1\frac{3}{4}

Decimal form:

w = - 1.75

w=-1.75

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
$| 4 w + 9 | = | 4 w + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 w + 9 \| = \| 4 w + 5 \|$
$x = + y$	$(4 w + 9) = (4 w + 5)$
$x = - y$	$(4 w + 9) = - (4 w + 5)$
$+ x = y$	$(4 w + 9) = (4 w + 5)$
$- x = y$	$- (4 w + 9) = (4 w + 5)$

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:

$\| x \| = \| y \|$	$\| 4 w + 9 \| = \| 4 w + 5 \|$
$x = + y$ , $+ x = y$	$(4 w + 9) = (4 w + 5)$
$x = - y$ , $- x = y$	$(4 w + 9) = - (4 w + 5)$

2. Solve the two equations for w

5 additional steps

$(4 w + 9) = (4 w + 5)$

Subtract from both sides:

$(4 w + 9) - 4 w = (4 w + 5) - 4 w$

Group like terms:

$(4 w - 4 w) + 9 = (4 w + 5) - 4 w$

Simplify the arithmetic:

$9 = (4 w + 5) - 4 w$

Group like terms:

$9 = (4 w - 4 w) + 5$

Simplify the arithmetic:

$9 = 5$

The statement is false:

$9 = 5$

The equation is false so it has no solution.

12 additional steps

$(4 w + 9) = - (4 w + 5)$

Expand the parentheses:

$(4 w + 9) = - 4 w - 5$

Add to both sides:

$(4 w + 9) + 4 w = (- 4 w - 5) + 4 w$

Group like terms:

$(4 w + 4 w) + 9 = (- 4 w - 5) + 4 w$

Simplify the arithmetic:

$8 w + 9 = (- 4 w - 5) + 4 w$

Group like terms:

$8 w + 9 = (- 4 w + 4 w) - 5$

Simplify the arithmetic:

$8 w + 9 = - 5$

Subtract from both sides:

$(8 w + 9) - 9 = - 5 - 9$

Simplify the arithmetic:

$8 w = - 5 - 9$

Simplify the arithmetic:

$8 w = - 14$

Divide both sides by :

$\frac{(8 w)}{8} = \frac{- 14}{8}$

Simplify the fraction:

$w = \frac{- 14}{8}$

Find the greatest common factor of the numerator and denominator:

$w = \frac{(- 7 \cdot 2)}{(4 \cdot 2)}$

Factor out and cancel the greatest common factor:

$w = \frac{- 7}{4}$

3. Graph

Each line represents the function of one side of the equation:
$y = | 4 w + 9 |$
$y = | 4 w + 5 |$
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