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Solution - Absolute value equations

Exact form:

b = 0, 0

b=0 , 0

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
$| - 2 b | = | 2 b |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 b \| = \| 2 b \|$
$x = + y$	$(- 2 b) = (2 b)$
$x = - y$	$(- 2 b) = - (2 b)$
$+ x = y$	$(- 2 b) = (2 b)$
$- x = y$	$- (- 2 b) = (2 b)$

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:

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

2. Solve the two equations for b

3 additional steps

$(- 2 b) = 2 b$

Subtract from both sides:

$(- 2 b) - 2 b = (2 b) - 2 b$

Simplify the arithmetic:

$- 4 b = (2 b) - 2 b$

Simplify the arithmetic:

$- 4 b = 0$

Divide both sides by the coefficient:

$b = 0$

7 additional steps

$(- 2 b) = - 2 b$

Divide both sides by :

$\frac{(- 2 b)}{- 2} = \frac{(- 2 b)}{- 2}$

Cancel out the negatives:

$\frac{2 b}{2} = \frac{(- 2 b)}{- 2}$

Simplify the fraction:

$b = \frac{(- 2 b)}{- 2}$

Cancel out the negatives:

$b = \frac{2 b}{2}$

Simplify the fraction:

$b = b$

Subtract from both sides:

$b - b = b - b$

Simplify the arithmetic:

$0 = b - b$

Simplify the arithmetic:

$0 = 0$

3. List the solutions

$b = 0, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 2 b |$
$y = | 2 b |$
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 b

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 b

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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