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Solution - Absolute value equations

Exact form:

b = - 2, 2

b=-2 , 2

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
$| b - 6 | = | 3 b - 2 |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for b

13 additional steps

$(b - 6) = (3 b - 2)$

Subtract from both sides:

$(b - 6) - 3 b = (3 b - 2) - 3 b$

Group like terms:

$(b - 3 b) - 6 = (3 b - 2) - 3 b$

Simplify the arithmetic:

$- 2 b - 6 = (3 b - 2) - 3 b$

Group like terms:

$- 2 b - 6 = (3 b - 3 b) - 2$

Simplify the arithmetic:

$- 2 b - 6 = - 2$

Add to both sides:

$(- 2 b - 6) + 6 = - 2 + 6$

Simplify the arithmetic:

$- 2 b = - 2 + 6$

Simplify the arithmetic:

$- 2 b = 4$

Divide both sides by :

$\frac{(- 2 b)}{- 2} = \frac{4}{- 2}$

Cancel out the negatives:

$\frac{2 b}{2} = \frac{4}{- 2}$

Simplify the fraction:

$b = \frac{4}{- 2}$

Move the negative sign from the denominator to the numerator:

$b = \frac{- 4}{2}$

Find the greatest common factor of the numerator and denominator:

$b = \frac{(- 2 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$b = - 2$

12 additional steps

$(b - 6) = - (3 b - 2)$

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 b - 6 = 2$

Add to both sides:

$(4 b - 6) + 6 = 2 + 6$

Simplify the arithmetic:

$4 b = 2 + 6$

Simplify the arithmetic:

$4 b = 8$

Divide both sides by :

$\frac{(4 b)}{4} = \frac{8}{4}$

Simplify the fraction:

$b = \frac{8}{4}$

Find the greatest common factor of the numerator and denominator:

$b = \frac{(2 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$b = 2$

3. List the solutions

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

4. Graph

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