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Solution - Absolute value equations

Exact form:

n = 0, 0

n=0 , 0

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
$| 36 n | = | 4 n |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 36 n \| = \| 4 n \|$
$x = + y$	$(36 n) = (4 n)$
$x = - y$	$(36 n) = - (4 n)$
$+ x = y$	$(36 n) = (4 n)$
$- x = y$	$- (36 n) = (4 n)$

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:

$\| x \| = \| y \|$	$\| 36 n \| = \| 4 n \|$
$x = + y$ , $+ x = y$	$(36 n) = (4 n)$
$x = - y$ , $- x = y$	$(36 n) = - (4 n)$

2. Solve the two equations for n

3 additional steps

$36 n = 4 n$

Subtract from both sides:

$(36 n) - 4 n = (4 n) - 4 n$

Simplify the arithmetic:

$32 n = (4 n) - 4 n$

Simplify the arithmetic:

$32 n = 0$

Divide both sides by the coefficient:

$n = 0$

12 additional steps

$36 n = - 4 n$

Divide both sides by :

$\frac{(36 n)}{36} = \frac{(- 4 n)}{36}$

Simplify the fraction:

$n = \frac{(- 4 n)}{36}$

Simplify the fraction:

$n = \frac{- 1}{9} n$

Add to both sides:

$n + \frac{1}{9} \cdot n = (\frac{- 1}{9} n) + \frac{1}{9} n$

Group the coefficients:

$(1 + \frac{1}{9}) n = (\frac{- 1}{9} \cdot n) + \frac{1}{9} n$

Convert the integer into a fraction:

$(\frac{9}{9} + \frac{1}{9}) n = (\frac{- 1}{9} \cdot n) + \frac{1}{9} n$

Combine the fractions:

$\frac{(9 + 1)}{9} \cdot n = (\frac{- 1}{9} \cdot n) + \frac{1}{9} n$

Combine the numerators:

$\frac{10}{9} \cdot n = (\frac{- 1}{9} \cdot n) + \frac{1}{9} n$

Combine the fractions:

$\frac{10}{9} \cdot n = \frac{(- 1 + 1)}{9} n$

Combine the numerators:

$\frac{10}{9} \cdot n = \frac{0}{9} n$

Reduce the zero numerator:

$\frac{10}{9} n = 0 n$

Simplify the arithmetic:

$\frac{10}{9} n = 0$

Divide both sides by the coefficient:

$n = 0$

3. List the solutions

$n = 0, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 36 n |$
$y = | 4 n |$
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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