Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{3 x - 1}{3 x - 1}$ .

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.2.3

Reorder terms.

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

Step 1.2.4

Rewrite $\frac{3 (3 x - 1) + 3}{3 x - 1}$ in a factored form.

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Step 1.2.4.1

Factor $3$ out of $3 (3 x - 1) + 3$ .

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Step 1.2.4.1.1

Factor $3$ out of $3$ .

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

Step 1.2.4.1.2

Factor $3$ out of $3 (3 x - 1) + 3 (1)$ .

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

Step 1.2.4.2

Add $- 1$ and $1$ .

$\frac{3}{3 x - 1} \cdot \frac{1}{\frac{3 (3 x + 0)}{3 x - 1}} - 3$

Step 1.2.4.3

Add $3 x$ and $0$ .

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

Step 1.2.4.4

Multiply $3$ by $3$ .

$\frac{3}{3 x - 1} \cdot \frac{1}{\frac{9 x}{3 x - 1}} - 3$

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{3 x - 1} (1 \frac{3 x - 1}{9 x}) - 3$

Step 1.4

Multiply $\frac{3 x - 1}{9 x}$ by $1$ .

$\frac{3}{3 x - 1} \cdot \frac{3 x - 1}{9 x} - 3$

Step 1.5

Cancel the common factor of $3$ .

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Step 1.5.1

Factor $3$ out of $9 x$ .

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

Step 1.5.2

Cancel the common factor.

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

Step 1.5.3

Rewrite the expression.

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

Step 1.6

Cancel the common factor of $3 x - 1$ .

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Step 1.6.1

Cancel the common factor.

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

Step 1.6.2

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Multiply $- 3$ by $3$ .

$\frac{1 - 9 x}{3 x}$