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Solution - Absolute value equations

Exact form:

n = \frac{5}{14}

n=\frac{5}{14}

Decimal form:

n = 0.357

n=0.357

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
$| 7 n - 8 | = | - 7 n - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 7 n - 8 \| = \| - 7 n - 3 \|$
$x = + y$	$(7 n - 8) = (- 7 n - 3)$
$x = - y$	$(7 n - 8) = - (- 7 n - 3)$
$+ x = y$	$(7 n - 8) = (- 7 n - 3)$
$- x = y$	$- (7 n - 8) = (- 7 n - 3)$

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:

$\| x \| = \| y \|$	$\| 7 n - 8 \| = \| - 7 n - 3 \|$
$x = + y$ , $+ x = y$	$(7 n - 8) = (- 7 n - 3)$
$x = - y$ , $- x = y$	$(7 n - 8) = - (- 7 n - 3)$

2. Solve the two equations for n

9 additional steps

$(7 n - 8) = (- 7 n - 3)$

Add to both sides:

$(7 n - 8) + 7 n = (- 7 n - 3) + 7 n$

Group like terms:

$(7 n + 7 n) - 8 = (- 7 n - 3) + 7 n$

Simplify the arithmetic:

$14 n - 8 = (- 7 n - 3) + 7 n$

Group like terms:

$14 n - 8 = (- 7 n + 7 n) - 3$

Simplify the arithmetic:

$14 n - 8 = - 3$

Add to both sides:

$(14 n - 8) + 8 = - 3 + 8$

Simplify the arithmetic:

$14 n = - 3 + 8$

Simplify the arithmetic:

$14 n = 5$

Divide both sides by :

$\frac{(14 n)}{14} = \frac{5}{14}$

Simplify the fraction:

$n = \frac{5}{14}$

6 additional steps

$(7 n - 8) = - (- 7 n - 3)$

Expand the parentheses:

$(7 n - 8) = 7 n + 3$

Subtract from both sides:

$(7 n - 8) - 7 n = (7 n + 3) - 7 n$

Group like terms:

$(7 n - 7 n) - 8 = (7 n + 3) - 7 n$

Simplify the arithmetic:

$- 8 = (7 n + 3) - 7 n$

Group like terms:

$- 8 = (7 n - 7 n) + 3$

Simplify the arithmetic:

$- 8 = 3$

The statement is false:

$- 8 = 3$

The equation is false so it has no solution.

3. List the solutions

$n = \frac{5}{14}$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 7 n - 8 |$
$y = | - 7 n - 3 |$
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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