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Solution - Absolute value equations

Exact form:

n = 2, 1

n=2 , 1

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
$| 3 n - 4 | = | n |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 n - 4 \| = \| n \|$
$x = + y$	$(3 n - 4) = (n)$
$x = - y$	$(3 n - 4) = - (n)$
$+ x = y$	$(3 n - 4) = (n)$
$- x = y$	$- (3 n - 4) = (n)$

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:

$\| x \| = \| y \|$	$\| 3 n - 4 \| = \| n \|$
$x = + y$ , $+ x = y$	$(3 n - 4) = (n)$
$x = - y$ , $- x = y$	$(3 n - 4) = - (n)$

2. Solve the two equations for n

10 additional steps

$(3 n - 4) = n$

Subtract from both sides:

$(3 n - 4) - n = n - n$

Group like terms:

$(3 n - n) - 4 = n - n$

Simplify the arithmetic:

$2 n - 4 = n - n$

Simplify the arithmetic:

$2 n - 4 = 0$

Add to both sides:

$(2 n - 4) + 4 = 0 + 4$

Simplify the arithmetic:

$2 n = 0 + 4$

Simplify the arithmetic:

$2 n = 4$

Divide both sides by :

$\frac{(2 n)}{2} = \frac{4}{2}$

Simplify the fraction:

$n = \frac{4}{2}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$n = 2$

9 additional steps

$(3 n - 4) = - n$

Add to both sides:

$(3 n - 4) + n = - n + n$

Group like terms:

$(3 n + n) - 4 = - n + n$

Simplify the arithmetic:

$4 n - 4 = - n + n$

Simplify the arithmetic:

$4 n - 4 = 0$

Add to both sides:

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

Simplify the arithmetic:

$4 n = 0 + 4$

Simplify the arithmetic:

$4 n = 4$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$n = 1$

3. List the solutions

$n = 2, 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 n - 4 |$
$y = | n |$
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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