Algebra Examples

$\frac{x + 1}{x - 4} - \frac{x + 1}{x^{2} - 7 x + 12}$

Step 1

Factor $x^{2} - 7 x + 12$ using the AC method.

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Step 1.1

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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 1.2

Write the factored form using these integers.

$\frac{x + 1}{x - 4} - \frac{x + 1}{(x - 4) (x - 3)}$

Step 2

To write $\frac{x + 1}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x + 1}{x - 4} \cdot \frac{x - 3}{x - 3} - \frac{x + 1}{(x - 4) (x - 3)}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{x + 1}{x - 4}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(x + 1) (x - 3)}{(x - 4) (x - 3)} - \frac{x + 1}{(x - 4) (x - 3)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{(x + 1) (x - 3) - (x + 1)}{(x - 4) (x - 3)}$

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

$\frac{(x + 1) (x - 3) + (x + 1) \cdot - 1}{(x - 4) (x - 3)}$

Step 4.1.2

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

$\frac{(x + 1) (x - 3 - 1)}{(x - 4) (x - 3)}$

Step 4.2

Subtract $1$ from $- 3$ .

$\frac{(x + 1) (x - 4)}{(x - 4) (x - 3)}$

Step 5

Cancel the common factor of $x - 4$ .

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Step 5.1

Cancel the common factor.

$\frac{(x + 1) (x - 4)}{(x - 4) (x - 3)}$

Step 5.2

Rewrite the expression.

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