Precalculus Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x}{x - 3} \cdot \frac{1}{\frac{2 x}{x - 3} + 2}$

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Rewrite $\frac{2 x + 2 (x - 3)}{x - 3}$ in a factored form.

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Step 2.3.1

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

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Step 2.3.1.1

Factor $2$ out of $2 x$ .

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

Step 2.3.1.2

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

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

Step 2.3.2

Add $x$ and $x$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Multiply $\frac{x - 3}{2 (2 x - 3)}$ by $1$ .

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

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Factor $2$ out of $2 x$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $x - 3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$x \frac{1}{2 x - 3}$

Step 7

Combine $x$ and $\frac{1}{2 x - 3}$ .

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