Calculus Examples

$- (\frac{7}{x - 2}) + \frac{1 - 3 x}{x}$

Step 1

To write $- \frac{7}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{7}{x - 2} \cdot \frac{x}{x} + \frac{1 - 3 x}{x}$

Step 2

To write $\frac{1 - 3 x}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$- \frac{7}{x - 2} \cdot \frac{x}{x} + \frac{1 - 3 x}{x} \cdot \frac{x - 2}{x - 2}$

Step 3

Write each expression with a common denominator of $(x - 2) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{7}{x - 2}$ by $\frac{x}{x}$ .

$- \frac{7 x}{(x - 2) x} + \frac{1 - 3 x}{x} \cdot \frac{x - 2}{x - 2}$

Step 3.2

Multiply $\frac{1 - 3 x}{x}$ by $\frac{x - 2}{x - 2}$ .

$- \frac{7 x}{(x - 2) x} + \frac{(1 - 3 x) (x - 2)}{x (x - 2)}$

Step 3.3

Reorder the factors of $(x - 2) x$ .

$- \frac{7 x}{x (x - 2)} + \frac{(1 - 3 x) (x - 2)}{x (x - 2)}$

Step 4

Combine the numerators over the common denominator.

$\frac{- 7 x + (1 - 3 x) (x - 2)}{x (x - 2)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(1 - 3 x) (x - 2)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{- 7 x + 1 (x - 2) - 3 x (x - 2)}{x (x - 2)}$

Step 5.1.2

Apply the distributive property.

$\frac{- 7 x + 1 x + 1 \cdot - 2 - 3 x (x - 2)}{x (x - 2)}$

Step 5.1.3

Apply the distributive property.

$\frac{- 7 x + 1 x + 1 \cdot - 2 - 3 x \cdot x - 3 x \cdot - 2}{x (x - 2)}$

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 $x$ by $1$ .

$\frac{- 7 x + x + 1 \cdot - 2 - 3 x \cdot x - 3 x \cdot - 2}{x (x - 2)}$

Step 5.2.1.2

Multiply $- 2$ by $1$ .

$\frac{- 7 x + x - 2 - 3 x \cdot x - 3 x \cdot - 2}{x (x - 2)}$

Step 5.2.1.3

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

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Step 5.2.1.3.1

Move $x$ .

$\frac{- 7 x + x - 2 - 3 (x \cdot x) - 3 x \cdot - 2}{x (x - 2)}$

Step 5.2.1.3.2

Multiply $x$ by $x$ .

$\frac{- 7 x + x - 2 - 3 x^{2} - 3 x \cdot - 2}{x (x - 2)}$

Step 5.2.1.4

Multiply $- 2$ by $- 3$ .

$\frac{- 7 x + x - 2 - 3 x^{2} + 6 x}{x (x - 2)}$

Step 5.2.2

Add $x$ and $6 x$ .

$\frac{- 7 x + 7 x - 2 - 3 x^{2}}{x (x - 2)}$

Step 5.3

Add $- 7 x$ and $7 x$ .

$\frac{0 - 2 - 3 x^{2}}{x (x - 2)}$

Step 5.4

Subtract $2$ from $0$ .

$\frac{- 2 - 3 x^{2}}{x (x - 2)}$

Step 6

Simplify with factoring out.

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Step 6.1

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- 1 (2) - 3 x^{2}}{x (x - 2)}$

Step 6.2

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

$\frac{- 1 (2) - (3 x^{2})}{x (x - 2)}$

Step 6.3

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

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

Step 6.4

Move the negative in front of the fraction.

$- \frac{2 + 3 x^{2}}{x (x - 2)}$