Precalculus Examples

$\frac{8}{3 x - 18} - \frac{x + 1}{x^{2} - 5 x - 6}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 x - 18$ .

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Step 1.1.1

Factor $3$ out of $3 x$ .

$\frac{8}{3 (x) - 18} - \frac{x + 1}{x^{2} - 5 x - 6}$

Step 1.1.2

Factor $3$ out of $- 18$ .

$\frac{8}{3 x + 3 \cdot - 6} - \frac{x + 1}{x^{2} - 5 x - 6}$

Step 1.1.3

Factor $3$ out of $3 x + 3 \cdot - 6$ .

$\frac{8}{3 (x - 6)} - \frac{x + 1}{x^{2} - 5 x - 6}$

Step 1.2

Factor $x^{2} - 5 x - 6$ using the AC method.

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Step 1.2.1

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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Step 1.2.2

Write the factored form using these integers.

$\frac{8}{3 (x - 6)} - \frac{x + 1}{(x - 6) (x + 1)}$

Step 1.3

Cancel the common factor of $x + 1$ .

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Step 1.3.1

Cancel the common factor.

$\frac{8}{3 (x - 6)} - \frac{x + 1}{(x - 6) (x + 1)}$

Step 1.3.2

Rewrite the expression.

$\frac{8}{3 (x - 6)} - \frac{1}{x - 6}$

Step 2

To write $- \frac{1}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3 (x - 6)} - \frac{1}{x - 6} \cdot \frac{3}{3}$

Step 3

Write each expression with a common denominator of $3 (x - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{x - 6}$ by $\frac{3}{3}$ .

$\frac{8}{3 (x - 6)} - \frac{3}{(x - 6) \cdot 3}$

Step 3.2

Reorder the factors of $(x - 6) \cdot 3$ .

$\frac{8}{3 (x - 6)} - \frac{3}{3 (x - 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{8 - 3}{3 (x - 6)}$

Step 5

Subtract $3$ from $8$ .

$\frac{5}{3 (x - 6)}$