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Solution - Absolute value equations

Exact form:

n = 3, - 3

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

$\| x \| = \| y \|$	$\| n - 3 \| = \| - n + 3 \|$
$x = + y$	$(n - 3) = (- n + 3)$
$x = - y$	$(n - 3) = - (- n + 3)$
$+ x = y$	$(n - 3) = (- n + 3)$
$- x = y$	$- (n - 3) = (- n + 3)$

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:

$\| x \| = \| y \|$	$\| n - 3 \| = \| - n + 3 \|$
$x = + y$ , $+ x = y$	$(n - 3) = (- n + 3)$
$x = - y$ , $- x = y$	$(n - 3) = - (- n + 3)$

2. Solve the two equations for n

11 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 n - 3 = 3$

Add to both sides:

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

Simplify the arithmetic:

$2 n = 3 + 3$

Simplify the arithmetic:

$2 n = 6$

Divide both sides by :

$\frac{(2 n)}{2} = \frac{6}{2}$

Simplify the fraction:

$n = \frac{6}{2}$

Find the greatest common factor of the numerator and denominator:

$n = \frac{(3 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$n = 3$

5 additional steps

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

Expand the parentheses:

$(n - 3) = n - 3$

Subtract from both sides:

$(n - 3) - n = (n - 3) - n$

Group like terms:

$(n - n) - 3 = (n - 3) - n$

Simplify the arithmetic:

$- 3 = (n - 3) - n$

Group like terms:

$- 3 = (n - n) - 3$

Simplify the arithmetic:

$- 3 = - 3$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | n - 3 |$
$y = | - n + 3 |$
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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