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Solution - Absolute value equations

Exact form:

w = - \frac{9}{4}

w=-\frac{9}{4}

Mixed number form:

w = - 2 \frac{1}{4}

w=-2\frac{1}{4}

Decimal form:

w = - 2.25

w=-2.25

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
$| 2 w + 6 | = | 2 w + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 w + 6 \| = \| 2 w + 3 \|$
$x = + y$	$(2 w + 6) = (2 w + 3)$
$x = - y$	$(2 w + 6) = - (2 w + 3)$
$+ x = y$	$(2 w + 6) = (2 w + 3)$
$- x = y$	$- (2 w + 6) = (2 w + 3)$

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:

$\| x \| = \| y \|$	$\| 2 w + 6 \| = \| 2 w + 3 \|$
$x = + y$ , $+ x = y$	$(2 w + 6) = (2 w + 3)$
$x = - y$ , $- x = y$	$(2 w + 6) = - (2 w + 3)$

2. Solve the two equations for w

5 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$6 = 3$

The statement is false:

$6 = 3$

The equation is false so it has no solution.

10 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$4 w + 6 = (- 2 w - 3) + 2 w$

Group like terms:

$4 w + 6 = (- 2 w + 2 w) - 3$

Simplify the arithmetic:

$4 w + 6 = - 3$

Subtract from both sides:

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

Simplify the arithmetic:

$4 w = - 3 - 6$

Simplify the arithmetic:

$4 w = - 9$

Divide both sides by :

$\frac{(4 w)}{4} = \frac{- 9}{4}$

Simplify the fraction:

$w = \frac{- 9}{4}$

3. Graph

Each line represents the function of one side of the equation:
$y = | 2 w + 6 |$
$y = | 2 w + 3 |$
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