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Solution - Absolute value equations

Exact form:

b = 2, \frac{2}{5}

b=2 , \frac{2}{5}

Decimal form:

b = 2, 0.4

b=2 , 0.4

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

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

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:

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

2. Solve the two equations for b

11 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$3 b - 4 = 2$

Add to both sides:

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

Simplify the arithmetic:

$3 b = 2 + 4$

Simplify the arithmetic:

$3 b = 6$

Divide both sides by :

$\frac{(3 b)}{3} = \frac{6}{3}$

Simplify the fraction:

$b = \frac{6}{3}$

Find the greatest common factor of the numerator and denominator:

$b = \frac{(2 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$b = 2$

10 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$5 b - 4 = (- b - 2) + b$

Group like terms:

$5 b - 4 = (- b + b) - 2$

Simplify the arithmetic:

$5 b - 4 = - 2$

Add to both sides:

$(5 b - 4) + 4 = - 2 + 4$

Simplify the arithmetic:

$5 b = - 2 + 4$

Simplify the arithmetic:

$5 b = 2$

Divide both sides by :

$\frac{(5 b)}{5} = \frac{2}{5}$

Simplify the fraction:

$b = \frac{2}{5}$

3. List the solutions

$b = 2, \frac{2}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 b - 4 |$
$y = | b + 2 |$
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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