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Solution - Absolute value equations

Exact form:

n = - 5, - 3

n=-5 , -3

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
$| 4 n + 15 | = | n |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 n + 15 \| = \| n \|$
$x = + y$	$(4 n + 15) = (n)$
$x = - y$	$(4 n + 15) = - (n)$
$+ x = y$	$(4 n + 15) = (n)$
$- x = y$	$- (4 n + 15) = (n)$

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:

$\| x \| = \| y \|$	$\| 4 n + 15 \| = \| n \|$
$x = + y$ , $+ x = y$	$(4 n + 15) = (n)$
$x = - y$ , $- x = y$	$(4 n + 15) = - (n)$

2. Solve the two equations for n

10 additional steps

$(4 n + 15) = n$

Subtract from both sides:

$(4 n + 15) - n = n - n$

Group like terms:

$(4 n - n) + 15 = n - n$

Simplify the arithmetic:

$3 n + 15 = n - n$

Simplify the arithmetic:

$3 n + 15 = 0$

Subtract from both sides:

$(3 n + 15) - 15 = 0 - 15$

Simplify the arithmetic:

$3 n = 0 - 15$

Simplify the arithmetic:

$3 n = - 15$

Divide both sides by :

$\frac{(3 n)}{3} = \frac{- 15}{3}$

Simplify the fraction:

$n = \frac{- 15}{3}$

Find the greatest common factor of the numerator and denominator:

$n = \frac{(- 5 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$n = - 5$

10 additional steps

$(4 n + 15) = - n$

Add to both sides:

$(4 n + 15) + n = - n + n$

Group like terms:

$(4 n + n) + 15 = - n + n$

Simplify the arithmetic:

$5 n + 15 = - n + n$

Simplify the arithmetic:

$5 n + 15 = 0$

Subtract from both sides:

$(5 n + 15) - 15 = 0 - 15$

Simplify the arithmetic:

$5 n = 0 - 15$

Simplify the arithmetic:

$5 n = - 15$

Divide both sides by :

$\frac{(5 n)}{5} = \frac{- 15}{5}$

Simplify the fraction:

$n = \frac{- 15}{5}$

Find the greatest common factor of the numerator and denominator:

$n = \frac{(- 3 \cdot 5)}{(1 \cdot 5)}$

Factor out and cancel the greatest common factor:

$n = - 3$

3. List the solutions

$n = - 5, - 3$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 n + 15 |$
$y = | n |$
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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