Precalculus Examples

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

Step 1

Simplify.

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Step 1.1

Divide $x - 3$ by $1$ .

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

Step 1.2

Divide $x$ by $1$ .

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

Step 2

Multiply the numerator and denominator of the fraction by $x - 2$ .

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Step 2.1

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

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 2$ .

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Step 4.1.1

Move the leading negative in $- \frac{3}{x - 2}$ into the numerator.

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Factor $x - 2$ out of $(x - 2) \cdot - 3$ .

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

Step 4.2.3

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

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

Step 5

Simplify the denominator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $x$ by $x$ .

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

Step 5.3

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 5.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 5.3.2

Write the factored form using these integers.

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

Step 6

Cancel the common factor of $x - 3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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