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Solution - Absolute value equations

Exact form:

b = 8

b=8

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
$| - b + 14 | = | - b + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - b + 14 \| = \| - b + 2 \|$
$x = + y$	$(- b + 14) = (- b + 2)$
$x = - y$	$(- b + 14) = - (- b + 2)$
$+ x = y$	$(- b + 14) = (- b + 2)$
$- x = y$	$- (- b + 14) = (- b + 2)$

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:

$\| x \| = \| y \|$	$\| - b + 14 \| = \| - b + 2 \|$
$x = + y$ , $+ x = y$	$(- b + 14) = (- b + 2)$
$x = - y$ , $- x = y$	$(- b + 14) = - (- b + 2)$

2. Solve the two equations for b

5 additional steps

$(- b + 14) = (- b + 2)$

Add to both sides:

$(- b + 14) + b = (- b + 2) + b$

Group like terms:

$(- b + b) + 14 = (- b + 2) + b$

Simplify the arithmetic:

$14 = (- b + 2) + b$

Group like terms:

$14 = (- b + b) + 2$

Simplify the arithmetic:

$14 = 2$

The statement is false:

$14 = 2$

The equation is false so it has no solution.

14 additional steps

$(- b + 14) = - (- b + 2)$

Expand the parentheses:

$(- b + 14) = b - 2$

Subtract from both sides:

$(- b + 14) - b = (b - 2) - b$

Group like terms:

$(- b - b) + 14 = (b - 2) - b$

Simplify the arithmetic:

$- 2 b + 14 = (b - 2) - b$

Group like terms:

$- 2 b + 14 = (b - b) - 2$

Simplify the arithmetic:

$- 2 b + 14 = - 2$

Subtract from both sides:

$(- 2 b + 14) - 14 = - 2 - 14$

Simplify the arithmetic:

$- 2 b = - 2 - 14$

Simplify the arithmetic:

$- 2 b = - 16$

Divide both sides by :

$\frac{(- 2 b)}{- 2} = \frac{- 16}{- 2}$

Cancel out the negatives:

$\frac{2 b}{2} = \frac{- 16}{- 2}$

Simplify the fraction:

$b = \frac{- 16}{- 2}$

Cancel out the negatives:

$b = \frac{16}{2}$

Find the greatest common factor of the numerator and denominator:

$b = \frac{(8 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$b = 8$

3. Graph

Each line represents the function of one side of the equation:
$y = | - b + 14 |$
$y = | - b + 2 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved