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Solution - Absolute value equations

Exact form:

n = - 3, - 2

n=-3 , -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
$| 5 n + 12 | = | n |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 n + 12 \| = \| n \|$
$x = + y$	$(5 n + 12) = (n)$
$x = - y$	$(5 n + 12) = - (n)$
$+ x = y$	$(5 n + 12) = (n)$
$- x = y$	$- (5 n + 12) = (n)$

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:

$\| x \| = \| y \|$	$\| 5 n + 12 \| = \| n \|$
$x = + y$ , $+ x = y$	$(5 n + 12) = (n)$
$x = - y$ , $- x = y$	$(5 n + 12) = - (n)$

2. Solve the two equations for n

10 additional steps

$(5 n + 12) = n$

Subtract from both sides:

$(5 n + 12) - n = n - n$

Group like terms:

$(5 n - n) + 12 = n - n$

Simplify the arithmetic:

$4 n + 12 = n - n$

Simplify the arithmetic:

$4 n + 12 = 0$

Subtract from both sides:

$(4 n + 12) - 12 = 0 - 12$

Simplify the arithmetic:

$4 n = 0 - 12$

Simplify the arithmetic:

$4 n = - 12$

Divide both sides by :

$\frac{(4 n)}{4} = \frac{- 12}{4}$

Simplify the fraction:

$n = \frac{- 12}{4}$

Find the greatest common factor of the numerator and denominator:

$n = \frac{(- 3 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$n = - 3$

10 additional steps

$(5 n + 12) = - n$

Add to both sides:

$(5 n + 12) + n = - n + n$

Group like terms:

$(5 n + n) + 12 = - n + n$

Simplify the arithmetic:

$6 n + 12 = - n + n$

Simplify the arithmetic:

$6 n + 12 = 0$

Subtract from both sides:

$(6 n + 12) - 12 = 0 - 12$

Simplify the arithmetic:

$6 n = 0 - 12$

Simplify the arithmetic:

$6 n = - 12$

Divide both sides by :

$\frac{(6 n)}{6} = \frac{- 12}{6}$

Simplify the fraction:

$n = \frac{- 12}{6}$

Find the greatest common factor of the numerator and denominator:

$n = \frac{(- 2 \cdot 6)}{(1 \cdot 6)}$

Factor out and cancel the greatest common factor:

$n = - 2$

3. List the solutions

$n = - 3, - 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 n + 12 |$
$y = | n |$
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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