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Solution - Absolute value equations

Exact form:

n = 15, 1

n=15 , 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
$| 2 n - 9 | = | n + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 n - 9 \| = \| n + 6 \|$
$x = + y$	$(2 n - 9) = (n + 6)$
$x = - y$	$(2 n - 9) = - (n + 6)$
$+ x = y$	$(2 n - 9) = (n + 6)$
$- x = y$	$- (2 n - 9) = (n + 6)$

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:

$\| x \| = \| y \|$	$\| 2 n - 9 \| = \| n + 6 \|$
$x = + y$ , $+ x = y$	$(2 n - 9) = (n + 6)$
$x = - y$ , $- x = y$	$(2 n - 9) = - (n + 6)$

2. Solve the two equations for n

7 additional steps

$(2 n - 9) = (n + 6)$

Subtract from both sides:

$(2 n - 9) - n = (n + 6) - n$

Group like terms:

$(2 n - n) - 9 = (n + 6) - n$

Simplify the arithmetic:

$n - 9 = (n + 6) - n$

Group like terms:

$n - 9 = (n - n) + 6$

Simplify the arithmetic:

$n - 9 = 6$

Add to both sides:

$(n - 9) + 9 = 6 + 9$

Simplify the arithmetic:

$n = 6 + 9$

Simplify the arithmetic:

$n = 15$

11 additional steps

$(2 n - 9) = - (n + 6)$

Expand the parentheses:

$(2 n - 9) = - n - 6$

Add to both sides:

$(2 n - 9) + n = (- n - 6) + n$

Group like terms:

$(2 n + n) - 9 = (- n - 6) + n$

Simplify the arithmetic:

$3 n - 9 = (- n - 6) + n$

Group like terms:

$3 n - 9 = (- n + n) - 6$

Simplify the arithmetic:

$3 n - 9 = - 6$

Add to both sides:

$(3 n - 9) + 9 = - 6 + 9$

Simplify the arithmetic:

$3 n = - 6 + 9$

Simplify the arithmetic:

$3 n = 3$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$n = 1$

3. List the solutions

$n = 15, 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 n - 9 |$
$y = | n + 6 |$
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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