Precalculus Examples

$- \frac{5}{x - 1} + \frac{1 - 3 x}{x}$

Step 1

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

$- \frac{5}{x - 1} \cdot \frac{x}{x} + \frac{1 - 3 x}{x}$

Step 2

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

$- \frac{5}{x - 1} \cdot \frac{x}{x} + \frac{1 - 3 x}{x} \cdot \frac{x - 1}{x - 1}$

Step 3

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

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Step 3.1

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

$- \frac{5 x}{(x - 1) x} + \frac{1 - 3 x}{x} \cdot \frac{x - 1}{x - 1}$

Step 3.2

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

$- \frac{5 x}{(x - 1) x} + \frac{(1 - 3 x) (x - 1)}{x (x - 1)}$

Step 3.3

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

$- \frac{5 x}{x (x - 1)} + \frac{(1 - 3 x) (x - 1)}{x (x - 1)}$

Step 4

Combine the numerators over the common denominator.

$\frac{- 5 x + (1 - 3 x) (x - 1)}{x (x - 1)}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

$\frac{- 5 x + 1 (x - 1) - 3 x (x - 1)}{x (x - 1)}$

Step 5.1.2

Apply the distributive property.

$\frac{- 5 x + 1 x + 1 \cdot - 1 - 3 x (x - 1)}{x (x - 1)}$

Step 5.1.3

Apply the distributive property.

$\frac{- 5 x + 1 x + 1 \cdot - 1 - 3 x \cdot x - 3 x \cdot - 1}{x (x - 1)}$

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 $1$ .

$\frac{- 5 x + x + 1 \cdot - 1 - 3 x \cdot x - 3 x \cdot - 1}{x (x - 1)}$

Step 5.2.1.2

Multiply $- 1$ by $1$ .

$\frac{- 5 x + x - 1 - 3 x \cdot x - 3 x \cdot - 1}{x (x - 1)}$

Step 5.2.1.3

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

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Step 5.2.1.3.1

Move $x$ .

$\frac{- 5 x + x - 1 - 3 (x \cdot x) - 3 x \cdot - 1}{x (x - 1)}$

Step 5.2.1.3.2

Multiply $x$ by $x$ .

$\frac{- 5 x + x - 1 - 3 x^{2} - 3 x \cdot - 1}{x (x - 1)}$

Step 5.2.1.4

Multiply $- 1$ by $- 3$ .

$\frac{- 5 x + x - 1 - 3 x^{2} + 3 x}{x (x - 1)}$

Step 5.2.2

Add $x$ and $3 x$ .

$\frac{- 5 x + 4 x - 1 - 3 x^{2}}{x (x - 1)}$

Step 5.3

Add $- 5 x$ and $4 x$ .

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

Step 5.4

Reorder terms.

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

Step 6

Simplify with factoring out.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

Simplify the expression.

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Step 6.6.1

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

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

Step 6.6.2

Move the negative in front of the fraction.

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