Finite Math Examples

$\frac{5 - x}{x + 1} - \frac{2 x - 1}{x - 6}$

Step 1

To write $\frac{5 - x}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{5 - x}{x + 1} \cdot \frac{x - 6}{x - 6} - \frac{2 x - 1}{x - 6}$

Step 2

To write $- \frac{2 x - 1}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{5 - x}{x + 1} \cdot \frac{x - 6}{x - 6} - \frac{2 x - 1}{x - 6} \cdot \frac{x + 1}{x + 1}$

Step 3

Write each expression with a common denominator of $(x + 1) (x - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5 - x}{x + 1}$ by $\frac{x - 6}{x - 6}$ .

$\frac{(5 - x) (x - 6)}{(x + 1) (x - 6)} - \frac{2 x - 1}{x - 6} \cdot \frac{x + 1}{x + 1}$

Step 3.2

Multiply $\frac{2 x - 1}{x - 6}$ by $\frac{x + 1}{x + 1}$ .

$\frac{(5 - x) (x - 6)}{(x + 1) (x - 6)} - \frac{(2 x - 1) (x + 1)}{(x - 6) (x + 1)}$

Step 3.3

Reorder the factors of $(x - 6) (x + 1)$ .

$\frac{(5 - x) (x - 6)}{(x + 1) (x - 6)} - \frac{(2 x - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(5 - x) (x - 6) - (2 x - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(5 - x) (x - 6)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{5 (x - 6) - x (x - 6) - (2 x - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.1.2

Apply the distributive property.

$\frac{5 x + 5 \cdot - 6 - x (x - 6) - (2 x - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.1.3

Apply the distributive property.

$\frac{5 x + 5 \cdot - 6 - x \cdot x - x \cdot - 6 - (2 x - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $5$ by $- 6$ .

$\frac{5 x - 30 - x \cdot x - x \cdot - 6 - (2 x - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.1.2.1

Move $x$ .

$\frac{5 x - 30 - (x \cdot x) - x \cdot - 6 - (2 x - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.2.1.2.2

Multiply $x$ by $x$ .

$\frac{5 x - 30 - x^{2} - x \cdot - 6 - (2 x - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.2.1.3

Multiply $- 6$ by $- 1$ .

$\frac{5 x - 30 - x^{2} + 6 x - (2 x - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.2.2

Add $5 x$ and $6 x$ .

$\frac{11 x - 30 - x^{2} - (2 x - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.3

Apply the distributive property.

$\frac{11 x - 30 - x^{2} + (- (2 x) - - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.4

Multiply $2$ by $- 1$ .

$\frac{11 x - 30 - x^{2} + (- 2 x - - 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.5

Multiply $- 1$ by $- 1$ .

$\frac{11 x - 30 - x^{2} + (- 2 x + 1) (x + 1)}{(x + 1) (x - 6)}$

Step 5.6

Expand $(- 2 x + 1) (x + 1)$ using the FOIL Method.

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Step 5.6.1

Apply the distributive property.

$\frac{11 x - 30 - x^{2} - 2 x (x + 1) + 1 (x + 1)}{(x + 1) (x - 6)}$

Step 5.6.2

Apply the distributive property.

$\frac{11 x - 30 - x^{2} - 2 x \cdot x - 2 x \cdot 1 + 1 (x + 1)}{(x + 1) (x - 6)}$

Step 5.6.3

Apply the distributive property.

$\frac{11 x - 30 - x^{2} - 2 x \cdot x - 2 x \cdot 1 + 1 x + 1 \cdot 1}{(x + 1) (x - 6)}$

Step 5.7

Simplify and combine like terms.

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Step 5.7.1

Simplify each term.

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Step 5.7.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.7.1.1.1

Move $x$ .

$\frac{11 x - 30 - x^{2} - 2 (x \cdot x) - 2 x \cdot 1 + 1 x + 1 \cdot 1}{(x + 1) (x - 6)}$

Step 5.7.1.1.2

Multiply $x$ by $x$ .

$\frac{11 x - 30 - x^{2} - 2 x^{2} - 2 x \cdot 1 + 1 x + 1 \cdot 1}{(x + 1) (x - 6)}$

Step 5.7.1.2

Multiply $- 2$ by $1$ .

$\frac{11 x - 30 - x^{2} - 2 x^{2} - 2 x + 1 x + 1 \cdot 1}{(x + 1) (x - 6)}$

Step 5.7.1.3

Multiply $x$ by $1$ .

$\frac{11 x - 30 - x^{2} - 2 x^{2} - 2 x + x + 1 \cdot 1}{(x + 1) (x - 6)}$

Step 5.7.1.4

Multiply $1$ by $1$ .

$\frac{11 x - 30 - x^{2} - 2 x^{2} - 2 x + x + 1}{(x + 1) (x - 6)}$

Step 5.7.2

Add $- 2 x$ and $x$ .

$\frac{11 x - 30 - x^{2} - 2 x^{2} - x + 1}{(x + 1) (x - 6)}$

Step 5.8

Subtract $x$ from $11 x$ .

$\frac{- 30 - x^{2} - 2 x^{2} + 10 x + 1}{(x + 1) (x - 6)}$

Step 5.9

Add $- 30$ and $1$ .

$\frac{- x^{2} - 2 x^{2} + 10 x - 29}{(x + 1) (x - 6)}$

Step 5.10

Subtract $2 x^{2}$ from $- x^{2}$ .

$\frac{- 3 x^{2} + 10 x - 29}{(x + 1) (x - 6)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 3 x^{2}$ .

$\frac{- (3 x^{2}) + 10 x - 29}{(x + 1) (x - 6)}$

Step 6.2

Factor $- 1$ out of $10 x$ .

$\frac{- (3 x^{2}) - (- 10 x) - 29}{(x + 1) (x - 6)}$

Step 6.3

Factor $- 1$ out of $- (3 x^{2}) - (- 10 x)$ .

$\frac{- (3 x^{2} - 10 x) - 29}{(x + 1) (x - 6)}$

Step 6.4

Rewrite $- 29$ as $- 1 (29)$ .

$\frac{- (3 x^{2} - 10 x) - 1 (29)}{(x + 1) (x - 6)}$

Step 6.5

Factor $- 1$ out of $- (3 x^{2} - 10 x) - 1 (29)$ .

$\frac{- (3 x^{2} - 10 x + 29)}{(x + 1) (x - 6)}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (3 x^{2} - 10 x + 29)$ as $- 1 (3 x^{2} - 10 x + 29)$ .

$\frac{- 1 (3 x^{2} - 10 x + 29)}{(x + 1) (x - 6)}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{3 x^{2} - 10 x + 29}{(x + 1) (x - 6)}$