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Solution - Absolute value equations

Exact form:

w = 5, 1

w=5 , 1

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
$| w + 1 | = | 2 w - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| w + 1 \| = \| 2 w - 4 \|$
$x = + y$	$(w + 1) = (2 w - 4)$
$x = - y$	$(w + 1) = - (2 w - 4)$
$+ x = y$	$(w + 1) = (2 w - 4)$
$- x = y$	$- (w + 1) = (2 w - 4)$

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:

$\| x \| = \| y \|$	$\| w + 1 \| = \| 2 w - 4 \|$
$x = + y$ , $+ x = y$	$(w + 1) = (2 w - 4)$
$x = - y$ , $- x = y$	$(w + 1) = - (2 w - 4)$

2. Solve the two equations for w

10 additional steps

$(w + 1) = (2 w - 4)$

Subtract from both sides:

$(w + 1) - 2 w = (2 w - 4) - 2 w$

Group like terms:

$(w - 2 w) + 1 = (2 w - 4) - 2 w$

Simplify the arithmetic:

$- w + 1 = (2 w - 4) - 2 w$

Group like terms:

$- w + 1 = (2 w - 2 w) - 4$

Simplify the arithmetic:

$- w + 1 = - 4$

Subtract from both sides:

$(- w + 1) - 1 = - 4 - 1$

Simplify the arithmetic:

$- w = - 4 - 1$

Simplify the arithmetic:

$- w = - 5$

Multiply both sides by :

$- w \cdot - 1 = - 5 \cdot - 1$

Remove the one(s):

$w = - 5 \cdot - 1$

Simplify the arithmetic:

$w = 5$

11 additional steps

$(w + 1) = - (2 w - 4)$

Expand the parentheses:

$(w + 1) = - 2 w + 4$

Add to both sides:

$(w + 1) + 2 w = (- 2 w + 4) + 2 w$

Group like terms:

$(w + 2 w) + 1 = (- 2 w + 4) + 2 w$

Simplify the arithmetic:

$3 w + 1 = (- 2 w + 4) + 2 w$

Group like terms:

$3 w + 1 = (- 2 w + 2 w) + 4$

Simplify the arithmetic:

$3 w + 1 = 4$

Subtract from both sides:

$(3 w + 1) - 1 = 4 - 1$

Simplify the arithmetic:

$3 w = 4 - 1$

Simplify the arithmetic:

$3 w = 3$

Divide both sides by :

$\frac{(3 w)}{3} = \frac{3}{3}$

Simplify the fraction:

$w = \frac{3}{3}$

Simplify the fraction:

$w = 1$

3. List the solutions

$w = 5, 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | w + 1 |$
$y = | 2 w - 4 |$
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. List the solutions

4. 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. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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