Pre-Algebra Examples

$\frac{7}{v + 5} + \frac{- 8 v - 5}{v^{2} + 5 v} - \frac{6 - v}{v}$

Step 1

Factor $v$ out of $v^{2} + 5 v$ .

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Step 1.1

Factor $v$ out of $v^{2}$ .

$\frac{7}{v + 5} + \frac{- 8 v - 5}{v \cdot v + 5 v} - \frac{6 - v}{v}$

Step 1.2

Factor $v$ out of $5 v$ .

$\frac{7}{v + 5} + \frac{- 8 v - 5}{v \cdot v + v \cdot 5} - \frac{6 - v}{v}$

Step 1.3

Factor $v$ out of $v \cdot v + v \cdot 5$ .

$\frac{7}{v + 5} + \frac{- 8 v - 5}{v (v + 5)} - \frac{6 - v}{v}$

Step 2

To write $\frac{7}{v + 5}$ as a fraction with a common denominator, multiply by $\frac{v}{v}$ .

$\frac{7}{v + 5} \cdot \frac{v}{v} + \frac{- 8 v - 5}{v (v + 5)} - \frac{6 - v}{v}$

Step 3

Write each expression with a common denominator of $(v + 5) v$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{7}{v + 5}$ by $\frac{v}{v}$ .

$\frac{7 v}{(v + 5) v} + \frac{- 8 v - 5}{v (v + 5)} - \frac{6 - v}{v}$

Step 3.2

Reorder the factors of $(v + 5) v$ .

$\frac{7 v}{v (v + 5)} + \frac{- 8 v - 5}{v (v + 5)} - \frac{6 - v}{v}$

Step 4

Simplify terms.

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Step 4.1

Combine the numerators over the common denominator.

$\frac{7 v - 8 v - 5}{v (v + 5)} - \frac{6 - v}{v}$

Step 4.2

Subtract $8 v$ from $7 v$ .

$\frac{- v - 5}{v (v + 5)} - \frac{6 - v}{v}$

Step 5

Simplify each term.

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Step 5.1

Cancel the common factor of $- v - 5$ and $v + 5$ .

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Step 5.1.1

Factor $- 1$ out of $- v$ .

$\frac{- (v) - 5}{v (v + 5)} - \frac{6 - v}{v}$

Step 5.1.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- (v) - 1 (5)}{v (v + 5)} - \frac{6 - v}{v}$

Step 5.1.3

Factor $- 1$ out of $- (v) - 1 (5)$ .

$\frac{- (v + 5)}{v (v + 5)} - \frac{6 - v}{v}$

Step 5.1.4

Rewrite $- (v + 5)$ as $- 1 (v + 5)$ .

$\frac{- 1 (v + 5)}{v (v + 5)} - \frac{6 - v}{v}$

Step 5.1.5

Cancel the common factor.

$\frac{- 1 (v + 5)}{v (v + 5)} - \frac{6 - v}{v}$

Step 5.1.6

Rewrite the expression.

$\frac{- 1}{v} - \frac{6 - v}{v}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{1}{v} - \frac{6 - v}{v}$

Step 6

Combine the numerators over the common denominator.

$\frac{- 1 - (6 - v)}{v}$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{- 1 - 1 \cdot 6 - - v}{v}$

Step 7.2

Multiply $- 1$ by $6$ .

$\frac{- 1 - 6 - - v}{v}$

Step 7.3

Multiply $- - v$ .

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Step 7.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- 1 - 6 + 1 v}{v}$

Step 7.3.2

Multiply $v$ by $1$ .

$\frac{- 1 - 6 + v}{v}$

Step 8

Simplify with factoring out.

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Step 8.1

Subtract $6$ from $- 1$ .

$\frac{- 7 + v}{v}$

Step 8.2

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{- 1 (7) + v}{v}$

Step 8.3

Factor $- 1$ out of $v$ .

$\frac{- 1 (7) - 1 (- v)}{v}$

Step 8.4

Factor $- 1$ out of $- 1 (7) - 1 (- v)$ .

$\frac{- 1 (7 - v)}{v}$

Step 8.5

Move the negative in front of the fraction.

$- \frac{7 - v}{v}$