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Solution - Absolute value equations

Exact form:

n = \frac{1}{3}, \frac{1}{5}

n=\frac{1}{3} , \frac{1}{5}

Decimal form:

n = 0.333, 0.2

n=0.333 , 0.2

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
$| 4 n - 1 | = | n |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 n - 1 \| = \| n \|$
$x = + y$	$(4 n - 1) = (n)$
$x = - y$	$(4 n - 1) = - (n)$
$+ x = y$	$(4 n - 1) = (n)$
$- x = y$	$- (4 n - 1) = (n)$

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:

$\| x \| = \| y \|$	$\| 4 n - 1 \| = \| n \|$
$x = + y$ , $+ x = y$	$(4 n - 1) = (n)$
$x = - y$ , $- x = y$	$(4 n - 1) = - (n)$

2. Solve the two equations for n

8 additional steps

$(4 n - 1) = n$

Subtract from both sides:

$(4 n - 1) - n = n - n$

Group like terms:

$(4 n - n) - 1 = n - n$

Simplify the arithmetic:

$3 n - 1 = n - n$

Simplify the arithmetic:

$3 n - 1 = 0$

Add to both sides:

$(3 n - 1) + 1 = 0 + 1$

Simplify the arithmetic:

$3 n = 0 + 1$

Simplify the arithmetic:

$3 n = 1$

Divide both sides by :

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

Simplify the fraction:

$n = \frac{1}{3}$

8 additional steps

$(4 n - 1) = - n$

Add to both sides:

$(4 n - 1) + n = - n + n$

Group like terms:

$(4 n + n) - 1 = - n + n$

Simplify the arithmetic:

$5 n - 1 = - n + n$

Simplify the arithmetic:

$5 n - 1 = 0$

Add to both sides:

$(5 n - 1) + 1 = 0 + 1$

Simplify the arithmetic:

$5 n = 0 + 1$

Simplify the arithmetic:

$5 n = 1$

Divide both sides by :

$\frac{(5 n)}{5} = \frac{1}{5}$

Simplify the fraction:

$n = \frac{1}{5}$

3. List the solutions

$n = \frac{1}{3}, \frac{1}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 n - 1 |$
$y = | n |$
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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