Algebra Examples

$2 | 4 - n | = - 3 n$

Step 1

Divide each term in $2 | 4 - n | = - 3 n$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 | 4 - n | = - 3 n$ by $2$ .

$\frac{2 | 4 - n |}{2} = \frac{- 3 n}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 | 4 - n |}{2} = \frac{- 3 n}{2}$

Step 1.2.1.2

Divide $| 4 - n |$ by $1$ .

$| 4 - n | = \frac{- 3 n}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$| 4 - n | = - \frac{3 n}{2}$

Step 2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$4 - n = \pm (- \frac{3 n}{2})$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$4 - n = - \frac{3 n}{2}$

Step 3.2

Multiply both sides by $2$ .

$(4 - n) \cdot 2 = - \frac{3 n}{2} \cdot 2$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $(4 - n) \cdot 2$ .

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Step 3.3.1.1.1

Apply the distributive property.

$4 \cdot 2 - n \cdot 2 = - \frac{3 n}{2} \cdot 2$

Step 3.3.1.1.2

Simplify the expression.

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Step 3.3.1.1.2.1

Multiply $4$ by $2$ .

$8 - n \cdot 2 = - \frac{3 n}{2} \cdot 2$

Step 3.3.1.1.2.2

Multiply $2$ by $- 1$ .

$8 - 2 n = - \frac{3 n}{2} \cdot 2$

Step 3.3.1.1.2.3

Reorder $8$ and $- 2 n$ .

$- 2 n + 8 = - \frac{3 n}{2} \cdot 2$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Move the leading negative in $- \frac{3 n}{2}$ into the numerator.

$- 2 n + 8 = \frac{- 3 n}{2} \cdot 2$

Step 3.3.2.1.2

Cancel the common factor.

$- 2 n + 8 = \frac{- 3 n}{2} \cdot 2$

Step 3.3.2.1.3

Rewrite the expression.

$- 2 n + 8 = - 3 n$

Step 3.4

Solve for $n$ .

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Step 3.4.1

Move all terms containing $n$ to the left side of the equation.

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Step 3.4.1.1

Add $3 n$ to both sides of the equation.

$- 2 n + 8 + 3 n = 0$

Step 3.4.1.2

Add $- 2 n$ and $3 n$ .

$n + 8 = 0$

Step 3.4.2

Subtract $8$ from both sides of the equation.

$n = - 8$

Step 3.5

Next, use the negative value of the $\pm$ to find the second solution.

$4 - n = \frac{3 n}{2}$

Step 3.6

Multiply both sides by $2$ .

$(4 - n) \cdot 2 = \frac{3 n}{2} \cdot 2$

Step 3.7

Simplify.

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Step 3.7.1

Simplify the left side.

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Step 3.7.1.1

Simplify $(4 - n) \cdot 2$ .

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Step 3.7.1.1.1

Apply the distributive property.

$4 \cdot 2 - n \cdot 2 = \frac{3 n}{2} \cdot 2$

Step 3.7.1.1.2

Simplify the expression.

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Step 3.7.1.1.2.1

Multiply $4$ by $2$ .

$8 - n \cdot 2 = \frac{3 n}{2} \cdot 2$

Step 3.7.1.1.2.2

Multiply $2$ by $- 1$ .

$8 - 2 n = \frac{3 n}{2} \cdot 2$

Step 3.7.1.1.2.3

Reorder $8$ and $- 2 n$ .

$- 2 n + 8 = \frac{3 n}{2} \cdot 2$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Cancel the common factor of $2$ .

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Step 3.7.2.1.1

Cancel the common factor.

$- 2 n + 8 = \frac{3 n}{2} \cdot 2$

Step 3.7.2.1.2

Rewrite the expression.

$- 2 n + 8 = 3 n$

Step 3.8

Solve for $n$ .

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Step 3.8.1

Move all terms containing $n$ to the left side of the equation.

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Step 3.8.1.1

Subtract $3 n$ from both sides of the equation.

$- 2 n + 8 - 3 n = 0$

Step 3.8.1.2

Subtract $3 n$ from $- 2 n$ .

$- 5 n + 8 = 0$

Step 3.8.2

Subtract $8$ from both sides of the equation.

$- 5 n = - 8$

Step 3.8.3

Divide each term in $- 5 n = - 8$ by $- 5$ and simplify.

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Step 3.8.3.1

Divide each term in $- 5 n = - 8$ by $- 5$ .

$\frac{- 5 n}{- 5} = \frac{- 8}{- 5}$

Step 3.8.3.2

Simplify the left side.

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Step 3.8.3.2.1

Cancel the common factor of $- 5$ .

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Step 3.8.3.2.1.1

Cancel the common factor.

$\frac{- 5 n}{- 5} = \frac{- 8}{- 5}$

Step 3.8.3.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 8}{- 5}$

Step 3.8.3.3

Simplify the right side.

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Step 3.8.3.3.1

Dividing two negative values results in a positive value.

$n = \frac{8}{5}$

Step 3.9

The complete solution is the result of both the positive and negative portions of the solution.

$n = - 8, \frac{8}{5}$

Step 4

Exclude the solutions that do not make $2 | 4 - n | = - 3 n$ true.

$n = - 8$