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Solution - Absolute value equations

Exact form:

n = 6, \frac{8}{3}

n=6 , \frac{8}{3}

Mixed number form:

n = 6, 2 \frac{2}{3}

n=6 , 2\frac{2}{3}

Decimal form:

n = 6, 2.667

n=6 , 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
$| n - 1 | = | 2 n - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| n - 1 \| = \| 2 n - 7 \|$
$x = + y$	$(n - 1) = (2 n - 7)$
$x = - y$	$(n - 1) = - (2 n - 7)$
$+ x = y$	$(n - 1) = (2 n - 7)$
$- x = y$	$- (n - 1) = (2 n - 7)$

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:

$\| x \| = \| y \|$	$\| n - 1 \| = \| 2 n - 7 \|$
$x = + y$ , $+ x = y$	$(n - 1) = (2 n - 7)$
$x = - y$ , $- x = y$	$(n - 1) = - (2 n - 7)$

2. Solve the two equations for n

10 additional steps

$(n - 1) = (2 n - 7)$

Subtract from both sides:

$(n - 1) - 2 n = (2 n - 7) - 2 n$

Group like terms:

$(n - 2 n) - 1 = (2 n - 7) - 2 n$

Simplify the arithmetic:

$- n - 1 = (2 n - 7) - 2 n$

Group like terms:

$- n - 1 = (2 n - 2 n) - 7$

Simplify the arithmetic:

$- n - 1 = - 7$

Add to both sides:

$(- n - 1) + 1 = - 7 + 1$

Simplify the arithmetic:

$- n = - 7 + 1$

Simplify the arithmetic:

$- n = - 6$

Multiply both sides by :

$- n \cdot - 1 = - 6 \cdot - 1$

Remove the one(s):

$n = - 6 \cdot - 1$

Simplify the arithmetic:

$n = 6$

10 additional steps

$(n - 1) = - (2 n - 7)$

Expand the parentheses:

$(n - 1) = - 2 n + 7$

Add to both sides:

$(n - 1) + 2 n = (- 2 n + 7) + 2 n$

Group like terms:

$(n + 2 n) - 1 = (- 2 n + 7) + 2 n$

Simplify the arithmetic:

$3 n - 1 = (- 2 n + 7) + 2 n$

Group like terms:

$3 n - 1 = (- 2 n + 2 n) + 7$

Simplify the arithmetic:

$3 n - 1 = 7$

Add to both sides:

$(3 n - 1) + 1 = 7 + 1$

Simplify the arithmetic:

$3 n = 7 + 1$

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

$n = 6, \frac{8}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | n - 1 |$
$y = | 2 n - 7 |$
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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