Precalculus Examples

$\frac{\frac{1}{x} - \frac{1}{2}}{x - 2}$

Step 1

Multiply the numerator and denominator of the fraction by $x \cdot 2$ .

Tap for more steps...

Step 1.1

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

$\frac{x \cdot 2}{x \cdot 2} \cdot \frac{\frac{1}{x} - \frac{1}{2}}{x - 2}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Tap for more steps...

Step 3.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.1.1

Factor $x$ out of $x \cdot 2$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 3.2.2

Factor $2$ out of $x \cdot 2$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Move $- 1$ to the left of $x$ .

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

Step 4.2

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

$\frac{2 - x}{x \cdot 2 x + x \cdot 2 \cdot - 2}$

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Move $x$ .

$\frac{2 - x}{x \cdot x \cdot 2 + x \cdot 2 \cdot - 2}$

Step 5.1.2

Multiply $x$ by $x$ .

$\frac{2 - x}{x^{2} \cdot 2 + x \cdot 2 \cdot - 2}$

Step 5.2

Move $2$ to the left of $x^{2}$ .

$\frac{2 - x}{2 \cdot x^{2} + x \cdot 2 \cdot - 2}$

Step 5.3

Move $2$ to the left of $x$ .

$\frac{2 - x}{2 x^{2} + 2 \cdot x \cdot - 2}$

Step 5.4

Multiply $- 2$ by $2$ .

$\frac{2 - x}{2 x^{2} - 4 x}$

Step 5.5

Factor $2 x$ out of $2 x^{2} - 4 x$ .

Tap for more steps...

Step 5.5.1

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

$\frac{2 - x}{2 x (x) - 4 x}$

Step 5.5.2

Factor $2 x$ out of $- 4 x$ .

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

Step 5.5.3

Factor $2 x$ out of $2 x (x) + 2 x (- 2)$ .

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

Step 6

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.1

Cancel the common factor of $2 - x$ and $x - 2$ .

Tap for more steps...

Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

Factor $- 1$ out of $- 1 (- 2) - (x)$ .

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

Step 6.1.4

Reorder terms.

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

Step 6.1.5

Cancel the common factor.

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

Step 6.1.6

Rewrite the expression.

$\frac{- 1}{2 x}$

Step 6.2

Move the negative in front of the fraction.

$- \frac{1}{2 x}$