Precalculus Examples

$\frac{3 x - 1}{x + 1} - \frac{x}{x - 1}$

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

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 $x$ by adding the exponents.

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Step 5.2.1.1.1

Move $x$ .

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

Step 5.2.1.1.2

Multiply $x$ by $x$ .

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

Step 5.2.1.2

Multiply $- 1$ by $3$ .

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

Step 5.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 5.2.1.4

Multiply $- 1$ by $- 1$ .

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

Step 5.2.2

Subtract $x$ from $- 3 x$ .

$\frac{3 x^{2} - 4 x + 1 - x (x + 1)}{(x + 1) (x - 1)}$

Step 5.3

Apply the distributive property.

$\frac{3 x^{2} - 4 x + 1 - x \cdot x - x \cdot 1}{(x + 1) (x - 1)}$

Step 5.4

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

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Step 5.4.1

Move $x$ .

$\frac{3 x^{2} - 4 x + 1 - (x \cdot x) - x \cdot 1}{(x + 1) (x - 1)}$

Step 5.4.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} - 4 x + 1 - x^{2} - x \cdot 1}{(x + 1) (x - 1)}$

Step 5.5

Multiply $- 1$ by $1$ .

$\frac{3 x^{2} - 4 x + 1 - x^{2} - x}{(x + 1) (x - 1)}$

Step 5.6

Subtract $x^{2}$ from $3 x^{2}$ .

$\frac{2 x^{2} - 4 x + 1 - x}{(x + 1) (x - 1)}$

Step 5.7

Subtract $x$ from $- 4 x$ .

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