Algebra Examples

$f (x) = \frac{2}{x - 3}$ $g (x) = \frac{6}{x}$

Step 1

Replace the function designators with the actual functions in $f (x) + g (x)$ .

$\frac{2}{x - 3} + \frac{6}{x}$

Step 2

Simplify.

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Step 2.1

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

$\frac{2}{x - 3} \cdot \frac{x}{x} + \frac{6}{x}$

Step 2.2

To write $\frac{6}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{2}{x - 3} \cdot \frac{x}{x} + \frac{6}{x} \cdot \frac{x - 3}{x - 3}$

Step 2.3

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

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Step 2.3.1

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

$\frac{2 x}{(x - 3) x} + \frac{6}{x} \cdot \frac{x - 3}{x - 3}$

Step 2.3.2

Multiply $\frac{6}{x}$ by $\frac{x - 3}{x - 3}$ .

$\frac{2 x}{(x - 3) x} + \frac{6 (x - 3)}{x (x - 3)}$

Step 2.3.3

Reorder the factors of $(x - 3) x$ .

$\frac{2 x}{x (x - 3)} + \frac{6 (x - 3)}{x (x - 3)}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{2 x + 6 (x - 3)}{x (x - 3)}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $2$ out of $2 x + 6 (x - 3)$ .

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Step 2.5.1.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 6 (x - 3)}{x (x - 3)}$

Step 2.5.1.2

Factor $2$ out of $6 (x - 3)$ .

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

Step 2.5.1.3

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

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Multiply $3$ by $- 3$ .

$\frac{2 (x + 3 x - 9)}{x (x - 3)}$

Step 2.5.4

Add $x$ and $3 x$ .

$\frac{2 (4 x - 9)}{x (x - 3)}$