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Solution - Absolute value equations

Exact form:

n = 3, - \frac{11}{3}

n=3 , -\frac{11}{3}

Mixed number form:

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

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

Decimal form:

n = 3, - 3.667

n=3 , -3.667

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
$| 4 n + 8 | = 2 | n + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 n + 8 \| = 2 \| n + 7 \|$
$x = + y$	$(4 n + 8) = 2 (n + 7)$
$x = - y$	$(4 n + 8) = 2 (- (n + 7))$
$+ x = y$	$(4 n + 8) = 2 (n + 7)$
$- x = y$	$- (4 n + 8) = 2 (n + 7)$

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:

$\| x \| = \| y \|$	$\| 4 n + 8 \| = 2 \| n + 7 \|$
$x = + y$ , $+ x = y$	$(4 n + 8) = 2 (n + 7)$
$x = - y$ , $- x = y$	$(4 n + 8) = 2 (- (n + 7))$

2. Solve the two equations for n

13 additional steps

$(4 n + 8) = 2 \cdot (n + 7)$

Expand the parentheses:

$(4 n + 8) = 2 n + 2 \cdot 7$

Simplify the arithmetic:

$(4 n + 8) = 2 n + 14$

Subtract from both sides:

$(4 n + 8) - 2 n = (2 n + 14) - 2 n$

Group like terms:

$(4 n - 2 n) + 8 = (2 n + 14) - 2 n$

Simplify the arithmetic:

$2 n + 8 = (2 n + 14) - 2 n$

Group like terms:

$2 n + 8 = (2 n - 2 n) + 14$

Simplify the arithmetic:

$2 n + 8 = 14$

Subtract from both sides:

$(2 n + 8) - 8 = 14 - 8$

Simplify the arithmetic:

$2 n = 14 - 8$

Simplify the arithmetic:

$2 n = 6$

Divide both sides by :

$\frac{(2 n)}{2} = \frac{6}{2}$

Simplify the fraction:

$n = \frac{6}{2}$

Find the greatest common factor of the numerator and denominator:

$n = \frac{(3 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$n = 3$

16 additional steps

$(4 n + 8) = 2 \cdot (- (n + 7))$

Expand the parentheses:

$(4 n + 8) = 2 \cdot (- n - 7)$

$(4 n + 8) = 2 \cdot - n + 2 \cdot - 7$

Group like terms:

$(4 n + 8) = (2 \cdot - 1) n + 2 \cdot - 7$

Multiply the coefficients:

$(4 n + 8) = - 2 n + 2 \cdot - 7$

Simplify the arithmetic:

$(4 n + 8) = - 2 n - 14$

Add to both sides:

$(4 n + 8) + 2 n = (- 2 n - 14) + 2 n$

Group like terms:

$(4 n + 2 n) + 8 = (- 2 n - 14) + 2 n$

Simplify the arithmetic:

$6 n + 8 = (- 2 n - 14) + 2 n$

Group like terms:

$6 n + 8 = (- 2 n + 2 n) - 14$

Simplify the arithmetic:

$6 n + 8 = - 14$

Subtract from both sides:

$(6 n + 8) - 8 = - 14 - 8$

Simplify the arithmetic:

$6 n = - 14 - 8$

Simplify the arithmetic:

$6 n = - 22$

Divide both sides by :

$\frac{(6 n)}{6} = \frac{- 22}{6}$

Simplify the fraction:

$n = \frac{- 22}{6}$

Find the greatest common factor of the numerator and denominator:

$n = \frac{(- 11 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$n = \frac{- 11}{3}$

3. List the solutions

$n = 3, - \frac{11}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 n + 8 |$
$y = 2 | n + 7 |$
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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