Calculus Examples

$\frac{2}{x - 2} - \frac{2}{3}$

Step 1

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

$\frac{2}{x - 2} \cdot \frac{3}{3} - \frac{2}{3}$

Step 2

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

$\frac{2}{x - 2} \cdot \frac{3}{3} - \frac{2}{3} \cdot \frac{x - 2}{x - 2}$

Step 3

Write each expression with a common denominator of $(x - 2) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(x - 2) \cdot 3$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Factor $2$ out of $2 \cdot 3 - 2 (x - 2)$ .

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Step 5.1.1

Factor $2$ out of $2 \cdot 3$ .

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

Step 5.1.2

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

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

Step 5.1.3

Factor $2$ out of $2 (3) + 2 (- (x - 2))$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $- 1$ by $- 2$ .

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

Step 5.4

Add $3$ and $2$ .

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

Step 6

Simplify with factoring out.

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Step 6.1

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

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

Step 6.2

Rewrite $5$ as $- 1 (- 5)$ .

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

Step 6.3

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

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

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

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

Step 6.4.2

Move the negative in front of the fraction.

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

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