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Solution - Absolute value equations

Exact form:

b = \frac{3}{2}, 3

b=\frac{3}{2} , 3

Mixed number form:

b = 1 \frac{1}{2}, 3

b=1\frac{1}{2} , 3

Decimal form:

b = 1.5, 3

b=1.5 , 3

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
$| - 2 b + 3 | = | 2 b - 3 |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for b

13 additional steps

$(- 2 b + 3) = (2 b - 3)$

Subtract from both sides:

$(- 2 b + 3) - 2 b = (2 b - 3) - 2 b$

Group like terms:

$(- 2 b - 2 b) + 3 = (2 b - 3) - 2 b$

Simplify the arithmetic:

$- 4 b + 3 = (2 b - 3) - 2 b$

Group like terms:

$- 4 b + 3 = (2 b - 2 b) - 3$

Simplify the arithmetic:

$- 4 b + 3 = - 3$

Subtract from both sides:

$(- 4 b + 3) - 3 = - 3 - 3$

Simplify the arithmetic:

$- 4 b = - 3 - 3$

Simplify the arithmetic:

$- 4 b = - 6$

Divide both sides by :

$\frac{(- 4 b)}{- 4} = \frac{- 6}{- 4}$

Cancel out the negatives:

$\frac{4 b}{4} = \frac{- 6}{- 4}$

Simplify the fraction:

$b = \frac{- 6}{- 4}$

Cancel out the negatives:

$b = \frac{6}{4}$

Find the greatest common factor of the numerator and denominator:

$b = \frac{(3 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$b = \frac{3}{2}$

5 additional steps

$(- 2 b + 3) = - (2 b - 3)$

Expand the parentheses:

$(- 2 b + 3) = - 2 b + 3$

Add to both sides:

$(- 2 b + 3) + 2 b = (- 2 b + 3) + 2 b$

Group like terms:

$(- 2 b + 2 b) + 3 = (- 2 b + 3) + 2 b$

Simplify the arithmetic:

$3 = (- 2 b + 3) + 2 b$

Group like terms:

$3 = (- 2 b + 2 b) + 3$

Simplify the arithmetic:

$3 = 3$

3. List the solutions

$b = \frac{3}{2}, 3$
(2 solution(s))

4. Graph

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