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Solution - Absolute value equations

Exact form:

n = 4, - \frac{7}{5}

n=4 , -\frac{7}{5}

Mixed number form:

n = 4, - 1 \frac{2}{5}

n=4 , -1\frac{2}{5}

Decimal form:

n = 4, - 1.4

n=4 , -1.4

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
$| 6 n + 3 | = | 4 n + 11 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 n + 3 \| = \| 4 n + 11 \|$
$x = + y$	$(6 n + 3) = (4 n + 11)$
$x = - y$	$(6 n + 3) = - (4 n + 11)$
$+ x = y$	$(6 n + 3) = (4 n + 11)$
$- x = y$	$- (6 n + 3) = (4 n + 11)$

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:

$\| x \| = \| y \|$	$\| 6 n + 3 \| = \| 4 n + 11 \|$
$x = + y$ , $+ x = y$	$(6 n + 3) = (4 n + 11)$
$x = - y$ , $- x = y$	$(6 n + 3) = - (4 n + 11)$

2. Solve the two equations for n

11 additional steps

$(6 n + 3) = (4 n + 11)$

Subtract from both sides:

$(6 n + 3) - 4 n = (4 n + 11) - 4 n$

Group like terms:

$(6 n - 4 n) + 3 = (4 n + 11) - 4 n$

Simplify the arithmetic:

$2 n + 3 = (4 n + 11) - 4 n$

Group like terms:

$2 n + 3 = (4 n - 4 n) + 11$

Simplify the arithmetic:

$2 n + 3 = 11$

Subtract from both sides:

$(2 n + 3) - 3 = 11 - 3$

Simplify the arithmetic:

$2 n = 11 - 3$

Simplify the arithmetic:

$2 n = 8$

Divide both sides by :

$\frac{(2 n)}{2} = \frac{8}{2}$

Simplify the fraction:

$n = \frac{8}{2}$

Find the greatest common factor of the numerator and denominator:

$n = \frac{(4 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$n = 4$

12 additional steps

$(6 n + 3) = - (4 n + 11)$

Expand the parentheses:

$(6 n + 3) = - 4 n - 11$

Add to both sides:

$(6 n + 3) + 4 n = (- 4 n - 11) + 4 n$

Group like terms:

$(6 n + 4 n) + 3 = (- 4 n - 11) + 4 n$

Simplify the arithmetic:

$10 n + 3 = (- 4 n - 11) + 4 n$

Group like terms:

$10 n + 3 = (- 4 n + 4 n) - 11$

Simplify the arithmetic:

$10 n + 3 = - 11$

Subtract from both sides:

$(10 n + 3) - 3 = - 11 - 3$

Simplify the arithmetic:

$10 n = - 11 - 3$

Simplify the arithmetic:

$10 n = - 14$

Divide both sides by :

$\frac{(10 n)}{10} = \frac{- 14}{10}$

Simplify the fraction:

$n = \frac{- 14}{10}$

Find the greatest common factor of the numerator and denominator:

$n = \frac{(- 7 \cdot 2)}{(5 \cdot 2)}$

Factor out and cancel the greatest common factor:

$n = \frac{- 7}{5}$

3. List the solutions

$n = 4, - \frac{7}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 n + 3 |$
$y = | 4 n + 11 |$
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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