Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(x - 1) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(x - 1) \cdot 3$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Move $x$ .

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

Step 5.2.2

Multiply $x$ by $x$ .

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

Step 5.3

Multiply $- x \cdot - 1$ .

Tap for more steps...

Step 5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.2

Multiply $x$ by $1$ .

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

Step 5.4

Reorder terms.

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

Step 6

Simplify with factoring out.

Tap for more steps...

Step 6.1

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

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

Step 6.2

Factor $- 1$ out of $x$ .

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

Step 6.3

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

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

Step 6.4

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 6.5

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

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

Step 6.6

Simplify the expression.

Tap for more steps...

Step 6.6.1

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

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

Step 6.6.2

Move the negative in front of the fraction.

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