Algebra Examples

$\frac{1}{x - 2} - \frac{5}{x + 7} = \frac{9}{x^{2} + 5 x - 14}$

Step 1

Subtract $\frac{9}{x^{2} + 5 x - 14}$ from both sides of the equation.

$\frac{1}{x - 2} - \frac{5}{x + 7} - \frac{9}{x^{2} + 5 x - 14} = 0$

Step 2

Simplify $\frac{1}{x - 2} - \frac{5}{x + 7} - \frac{9}{x^{2} + 5 x - 14}$ .

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Step 2.1

Factor $x^{2} + 5 x - 14$ using the AC method.

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Step 2.1.1

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 $- 14$ and whose sum is $5$ .

$- 2, 7$

Step 2.1.2

Write the factored form using these integers.

$\frac{1}{x - 2} - \frac{5}{x + 7} - \frac{9}{(x - 2) (x + 7)} = 0$

Step 2.2

To write $\frac{1}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$\frac{1}{x - 2} \cdot \frac{x + 7}{x + 7} - \frac{5}{x + 7} - \frac{9}{(x - 2) (x + 7)} = 0$

Step 2.3

To write $- \frac{5}{x + 7}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{1}{x - 2} \cdot \frac{x + 7}{x + 7} - \frac{5}{x + 7} \cdot \frac{x - 2}{x - 2} - \frac{9}{(x - 2) (x + 7)} = 0$

Step 2.4

Write each expression with a common denominator of $(x - 2) (x + 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{1}{x - 2}$ by $\frac{x + 7}{x + 7}$ .

$\frac{x + 7}{(x - 2) (x + 7)} - \frac{5}{x + 7} \cdot \frac{x - 2}{x - 2} - \frac{9}{(x - 2) (x + 7)} = 0$

Step 2.4.2

Multiply $\frac{5}{x + 7}$ by $\frac{x - 2}{x - 2}$ .

$\frac{x + 7}{(x - 2) (x + 7)} - \frac{5 (x - 2)}{(x + 7) (x - 2)} - \frac{9}{(x - 2) (x + 7)} = 0$

Step 2.4.3

Reorder the factors of $(x - 2) (x + 7)$ .

$\frac{x + 7}{(x + 7) (x - 2)} - \frac{5 (x - 2)}{(x + 7) (x - 2)} - \frac{9}{(x - 2) (x + 7)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{x + 7 - 5 (x - 2)}{(x + 7) (x - 2)} - \frac{9}{(x - 2) (x + 7)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{x + 7 - 5 (x - 2) - 9}{(x + 7) (x - 2)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

$\frac{x + 7 - 5 x - 5 \cdot - 2 - 9}{(x + 7) (x - 2)} = 0$

Step 2.7.2

Multiply $- 5$ by $- 2$ .

$\frac{x + 7 - 5 x + 10 - 9}{(x + 7) (x - 2)} = 0$

Step 2.8

Subtract $5 x$ from $x$ .

$\frac{- 4 x + 7 + 10 - 9}{(x + 7) (x - 2)} = 0$

Step 2.9

Add $7$ and $10$ .

$\frac{- 4 x + 17 - 9}{(x + 7) (x - 2)} = 0$

Step 2.10

Subtract $9$ from $17$ .

$\frac{- 4 x + 8}{(x + 7) (x - 2)} = 0$

Step 2.11

Factor $4$ out of $- 4 x + 8$ .

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Step 2.11.1

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) + 8}{(x + 7) (x - 2)} = 0$

Step 2.11.2

Factor $4$ out of $8$ .

$\frac{4 (- x) + 4 (2)}{(x + 7) (x - 2)} = 0$

Step 2.11.3

Factor $4$ out of $4 (- x) + 4 (2)$ .

$\frac{4 (- x + 2)}{(x + 7) (x - 2)} = 0$

Step 2.12

Cancel the common factor of $- x + 2$ and $x - 2$ .

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Step 2.12.1

Factor $- 1$ out of $- x$ .

$\frac{4 (- (x) + 2)}{(x + 7) (x - 2)} = 0$

Step 2.12.2

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{4 (- (x) - 1 (- 2))}{(x + 7) (x - 2)} = 0$

Step 2.12.3

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$\frac{4 (- (x - 2))}{(x + 7) (x - 2)} = 0$

Step 2.12.4

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$\frac{4 (- 1 (x - 2))}{(x + 7) (x - 2)} = 0$

Step 2.12.5

Cancel the common factor.

$\frac{4 (- 1 (x - 2))}{(x + 7) (x - 2)} = 0$

Step 2.12.6

Rewrite the expression.

$\frac{4 \cdot (- 1)}{x + 7} = 0$

Step 2.13

Multiply $4$ by $- 1$ .

$\frac{- 4}{x + 7} = 0$

Step 2.14

Move the negative in front of the fraction.

$- \frac{4}{x + 7} = 0$

Step 3

Set the denominator in $\frac{4}{x + 7}$ equal to $0$ to find where the expression is undefined.

$x + 7 = 0$

Step 4

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 5