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Solution - Absolute value equations

Exact form:

n = \frac{15}{13}, 1

n=\frac{15}{13} , 1

Mixed number form:

n = 1 \frac{2}{13}, 1

n=1\frac{2}{13} , 1

Decimal form:

n = 1.154, 1

n=1.154 , 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
$| 14 n - 15 | = | n |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 14 n - 15 \| = \| n \|$
$x = + y$	$(14 n - 15) = (n)$
$x = - y$	$(14 n - 15) = - (n)$
$+ x = y$	$(14 n - 15) = (n)$
$- x = y$	$- (14 n - 15) = (n)$

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:

$\| x \| = \| y \|$	$\| 14 n - 15 \| = \| n \|$
$x = + y$ , $+ x = y$	$(14 n - 15) = (n)$
$x = - y$ , $- x = y$	$(14 n - 15) = - (n)$

2. Solve the two equations for n

8 additional steps

$(14 n - 15) = n$

Subtract from both sides:

$(14 n - 15) - n = n - n$

Group like terms:

$(14 n - n) - 15 = n - n$

Simplify the arithmetic:

$13 n - 15 = n - n$

Simplify the arithmetic:

$13 n - 15 = 0$

Add to both sides:

$(13 n - 15) + 15 = 0 + 15$

Simplify the arithmetic:

$13 n = 0 + 15$

Simplify the arithmetic:

$13 n = 15$

Divide both sides by :

$\frac{(13 n)}{13} = \frac{15}{13}$

Simplify the fraction:

$n = \frac{15}{13}$

9 additional steps

$(14 n - 15) = - n$

Add to both sides:

$(14 n - 15) + n = - n + n$

Group like terms:

$(14 n + n) - 15 = - n + n$

Simplify the arithmetic:

$15 n - 15 = - n + n$

Simplify the arithmetic:

$15 n - 15 = 0$

Add to both sides:

$(15 n - 15) + 15 = 0 + 15$

Simplify the arithmetic:

$15 n = 0 + 15$

Simplify the arithmetic:

$15 n = 15$

Divide both sides by :

$\frac{(15 n)}{15} = \frac{15}{15}$

Simplify the fraction:

$n = \frac{15}{15}$

Simplify the fraction:

$n = 1$

3. List the solutions

$n = \frac{15}{13}, 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 14 n - 15 |$
$y = | n |$
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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