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Solution - Absolute value equations

Exact form:

n = 4, \frac{8}{3}

n=4 , \frac{8}{3}

Mixed number form:

n = 4, 2 \frac{2}{3}

n=4 , 2\frac{2}{3}

Decimal form:

n = 4, 2.667

n=4 , 2.667

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
$| n - 2 | = 2 | n - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| n - 2 \| = 2 \| n - 3 \|$
$x = + y$	$(n - 2) = 2 (n - 3)$
$x = - y$	$(n - 2) = 2 (- (n - 3))$
$+ x = y$	$(n - 2) = 2 (n - 3)$
$- x = y$	$- (n - 2) = 2 (n - 3)$

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:

$\| x \| = \| y \|$	$\| n - 2 \| = 2 \| n - 3 \|$
$x = + y$ , $+ x = y$	$(n - 2) = 2 (n - 3)$
$x = - y$ , $- x = y$	$(n - 2) = 2 (- (n - 3))$

2. Solve the two equations for n

12 additional steps

$(n - 2) = 2 \cdot (n - 3)$

Expand the parentheses:

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

Simplify the arithmetic:

$(n - 2) = 2 n - 6$

Subtract from both sides:

$(n - 2) - 2 n = (2 n - 6) - 2 n$

Group like terms:

$(n - 2 n) - 2 = (2 n - 6) - 2 n$

Simplify the arithmetic:

$- n - 2 = (2 n - 6) - 2 n$

Group like terms:

$- n - 2 = (2 n - 2 n) - 6$

Simplify the arithmetic:

$- n - 2 = - 6$

Add to both sides:

$(- n - 2) + 2 = - 6 + 2$

Simplify the arithmetic:

$- n = - 6 + 2$

Simplify the arithmetic:

$- n = - 4$

Multiply both sides by :

$- n \cdot - 1 = - 4 \cdot - 1$

Remove the one(s):

$n = - 4 \cdot - 1$

Simplify the arithmetic:

$n = 4$

14 additional steps

$(n - 2) = 2 \cdot (- (n - 3))$

Expand the parentheses:

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

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

Group like terms:

$(n - 2) = (2 \cdot - 1) n + 2 \cdot 3$

Multiply the coefficients:

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

Simplify the arithmetic:

$(n - 2) = - 2 n + 6$

Add to both sides:

$(n - 2) + 2 n = (- 2 n + 6) + 2 n$

Group like terms:

$(n + 2 n) - 2 = (- 2 n + 6) + 2 n$

Simplify the arithmetic:

$3 n - 2 = (- 2 n + 6) + 2 n$

Group like terms:

$3 n - 2 = (- 2 n + 2 n) + 6$

Simplify the arithmetic:

$3 n - 2 = 6$

Add to both sides:

$(3 n - 2) + 2 = 6 + 2$

Simplify the arithmetic:

$3 n = 6 + 2$

Simplify the arithmetic:

$3 n = 8$

Divide both sides by :

$\frac{(3 n)}{3} = \frac{8}{3}$

Simplify the fraction:

$n = \frac{8}{3}$

3. List the solutions

$n = 4, \frac{8}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | n - 2 |$
$y = 2 | n - 3 |$
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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