Precalculus Examples

$\frac{1}{(\frac{1}{2 x} - 3) + 6}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{1}{\frac{1}{2 x} - 3 \cdot \frac{2 x}{2 x} + 6}$

Step 1.2

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

$\frac{1}{\frac{1}{2 x} + \frac{- 3 (2 x)}{2 x} + 6}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{1}{\frac{1 - 3 (2 x)}{2 x} + 6}$

Step 1.4

To write $6$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{1}{\frac{1 - 3 \cdot 2 x}{2 x} + 6 \cdot \frac{2 x}{2 x}}$

Step 1.5

Combine $6$ and $\frac{2 x}{2 x}$ .

$\frac{1}{\frac{1 - 3 \cdot 2 x}{2 x} + \frac{6 (2 x)}{2 x}}$

Step 1.6

Combine the numerators over the common denominator.

$\frac{1}{\frac{1 - 3 \cdot 2 x + 6 (2 x)}{2 x}}$

Step 1.7

Reorder terms.

$\frac{1}{\frac{- 3 \cdot 2 x + 2 \cdot 6 x + 1}{2 x}}$

Step 1.8

Rewrite $\frac{- 3 \cdot 2 x + 2 \cdot 6 x + 1}{2 x}$ in a factored form.

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Step 1.8.1

Multiply $- 3$ by $2$ .

$\frac{1}{\frac{- 6 x + 2 \cdot 6 x + 1}{2 x}}$

Step 1.8.2

Multiply $2$ by $6$ .

$\frac{1}{\frac{- 6 x + 12 x + 1}{2 x}}$

Step 1.8.3

Add $- 6 x$ and $12 x$ .

$\frac{1}{\frac{6 x + 1}{2 x}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{2 x}{6 x + 1}$

Step 3

Multiply $\frac{2 x}{6 x + 1}$ by $1$ .

$\frac{2 x}{6 x + 1}$