Algebra Examples

$f (\frac{5}{x}) = \frac{6 (\frac{5}{x}) + 2}{4 (\frac{5}{x}) + 3}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2$ out of $6 \frac{5}{x} + 2$ .

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Step 1.1.1

Factor $2$ out of $6 \frac{5}{x}$ .

$f (\frac{5}{x}) = \frac{2 (3 (\frac{5}{x})) + 2}{4 (\frac{5}{x}) + 3}$

Step 1.1.2

Factor $2$ out of $2$ .

$f (\frac{5}{x}) = \frac{2 (3 (\frac{5}{x})) + 2 (1)}{4 (\frac{5}{x}) + 3}$

Step 1.1.3

Factor $2$ out of $2 (3 \frac{5}{x}) + 2 (1)$ .

$f (\frac{5}{x}) = \frac{2 (3 (\frac{5}{x}) + 1)}{4 (\frac{5}{x}) + 3}$

Step 1.2

Multiply $3 \frac{5}{x}$ .

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Step 1.2.1

Combine $3$ and $\frac{5}{x}$ .

$f (\frac{5}{x}) = \frac{2 (\frac{3 \cdot 5}{x} + 1)}{4 (\frac{5}{x}) + 3}$

Step 1.2.2

Multiply $3$ by $5$ .

$f (\frac{5}{x}) = \frac{2 (\frac{15}{x} + 1)}{4 (\frac{5}{x}) + 3}$

Step 1.3

Write $1$ as a fraction with a common denominator.

$f (\frac{5}{x}) = \frac{2 (\frac{15}{x} + \frac{x}{x})}{4 (\frac{5}{x}) + 3}$

Step 1.4

Combine the numerators over the common denominator.

$f (\frac{5}{x}) = \frac{2 (\frac{15 + x}{x})}{4 (\frac{5}{x}) + 3}$

Step 2

Simplify the denominator.

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Step 2.1

Multiply $4 \frac{5}{x}$ .

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Step 2.1.1

Combine $4$ and $\frac{5}{x}$ .

$f (\frac{5}{x}) = \frac{2 (\frac{15 + x}{x})}{\frac{4 \cdot 5}{x} + 3}$

Step 2.1.2

Multiply $4$ by $5$ .

$f (\frac{5}{x}) = \frac{2 (\frac{15 + x}{x})}{\frac{20}{x} + 3}$

Step 2.2

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

$f (\frac{5}{x}) = \frac{2 (\frac{15 + x}{x})}{\frac{20}{x} + \frac{3 x}{x}}$

Step 2.3

Combine the numerators over the common denominator.

$f (\frac{5}{x}) = \frac{2 (\frac{15 + x}{x})}{\frac{20 + 3 x}{x}}$

Step 3

Combine $2$ and $\frac{15 + x}{x}$ .

$f (\frac{5}{x}) = \frac{\frac{2 (15 + x)}{x}}{\frac{20 + 3 x}{x}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5}{x}) = \frac{2 (15 + x)}{x} \cdot \frac{x}{20 + 3 x}$

Step 5

Cancel the common factor of $x$ .

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Step 5.1

Cancel the common factor.

$f (\frac{5}{x}) = \frac{2 (15 + x)}{x} \cdot \frac{x}{20 + 3 x}$

Step 5.2

Rewrite the expression.

$f (\frac{5}{x}) = 2 (15 + x) (\frac{1}{20 + 3 x})$

Step 6

Combine $\frac{1}{20 + 3 x}$ and $2$ .

$f (\frac{5}{x}) = \frac{2}{20 + 3 x} \cdot (15 + x)$

Step 7

Multiply $\frac{2}{20 + 3 x}$ by $15 + x$ .

$f (\frac{5}{x}) = \frac{2 (15 + x)}{20 + 3 x}$

Step 8