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Solution - Absolute value equations

Exact form:

n = - 14, - 6

n=-14 , -6

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
$| n + 2 | = 2 | n + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| n + 2 \| = 2 \| n + 8 \|$
$x = + y$	$(n + 2) = 2 (n + 8)$
$x = - y$	$(n + 2) = 2 (- (n + 8))$
$+ x = y$	$(n + 2) = 2 (n + 8)$
$- x = y$	$- (n + 2) = 2 (n + 8)$

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:

$\| x \| = \| y \|$	$\| n + 2 \| = 2 \| n + 8 \|$
$x = + y$ , $+ x = y$	$(n + 2) = 2 (n + 8)$
$x = - y$ , $- x = y$	$(n + 2) = 2 (- (n + 8))$

2. Solve the two equations for n

12 additional steps

$(n + 2) = 2 \cdot (n + 8)$

Expand the parentheses:

$(n + 2) = 2 n + 2 \cdot 8$

Simplify the arithmetic:

$(n + 2) = 2 n + 16$

Subtract from both sides:

$(n + 2) - 2 n = (2 n + 16) - 2 n$

Group like terms:

$(n - 2 n) + 2 = (2 n + 16) - 2 n$

Simplify the arithmetic:

$- n + 2 = (2 n + 16) - 2 n$

Group like terms:

$- n + 2 = (2 n - 2 n) + 16$

Simplify the arithmetic:

$- n + 2 = 16$

Subtract from both sides:

$(- n + 2) - 2 = 16 - 2$

Simplify the arithmetic:

$- n = 16 - 2$

Simplify the arithmetic:

$- n = 14$

Multiply both sides by :

$- n \cdot - 1 = 14 \cdot - 1$

Remove the one(s):

$n = 14 \cdot - 1$

Simplify the arithmetic:

$n = - 14$

16 additional steps

$(n + 2) = 2 \cdot (- (n + 8))$

Expand the parentheses:

$(n + 2) = 2 \cdot (- n - 8)$

$(n + 2) = 2 \cdot - n + 2 \cdot - 8$

Group like terms:

$(n + 2) = (2 \cdot - 1) n + 2 \cdot - 8$

Multiply the coefficients:

$(n + 2) = - 2 n + 2 \cdot - 8$

Simplify the arithmetic:

$(n + 2) = - 2 n - 16$

Add to both sides:

$(n + 2) + 2 n = (- 2 n - 16) + 2 n$

Group like terms:

$(n + 2 n) + 2 = (- 2 n - 16) + 2 n$

Simplify the arithmetic:

$3 n + 2 = (- 2 n - 16) + 2 n$

Group like terms:

$3 n + 2 = (- 2 n + 2 n) - 16$

Simplify the arithmetic:

$3 n + 2 = - 16$

Subtract from both sides:

$(3 n + 2) - 2 = - 16 - 2$

Simplify the arithmetic:

$3 n = - 16 - 2$

Simplify the arithmetic:

$3 n = - 18$

Divide both sides by :

$\frac{(3 n)}{3} = \frac{- 18}{3}$

Simplify the fraction:

$n = \frac{- 18}{3}$

Find the greatest common factor of the numerator and denominator:

$n = \frac{(- 6 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$n = - 6$

3. List the solutions

$n = - 14, - 6$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | n + 2 |$
$y = 2 | n + 8 |$
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 n

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 n

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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