Find Where Undefined/Discontinuous 1/(x-2)-3/(x+7)=9/(x^2+5x-14)

$\frac{1}{x - 2} - \frac{3}{x + 7} = \frac{9}{x^{2} + 5 x - 14}$

Step 1

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

$\frac{1}{x - 2} - \frac{3}{x + 7} - \frac{9}{x^{2} + 5 x - 14} = 0$

Step 2

Simplify $\frac{1}{x - 2} - \frac{3}{x + 7} - \frac{9}{x^{2} + 5 x - 14}$ .

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

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 $5$ .

$- 2, 7$

Write the factored form using these integers.

$\frac{1}{x - 2} - \frac{3}{x + 7} - \frac{9}{(x - 2) (x + 7)} = 0$

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

$\frac{1}{x - 2} \cdot \frac{x + 7}{x + 7} - \frac{3}{x + 7} - \frac{9}{(x - 2) (x + 7)} = 0$

To write $- \frac{3}{x + 7}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{1}{x - 2} \cdot \frac{x + 7}{x + 7} - \frac{3}{x + 7} \cdot \frac{x - 2}{x - 2} - \frac{9}{(x - 2) (x + 7)} = 0$

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

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

$\frac{x + 7}{(x - 2) (x + 7)} - \frac{3}{x + 7} \cdot \frac{x - 2}{x - 2} - \frac{9}{(x - 2) (x + 7)} = 0$

Multiply $\frac{3}{x + 7}$ by $\frac{x - 2}{x - 2}$ .

$\frac{x + 7}{(x - 2) (x + 7)} - \frac{3 (x - 2)}{(x + 7) (x - 2)} - \frac{9}{(x - 2) (x + 7)} = 0$

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

$\frac{x + 7}{(x + 7) (x - 2)} - \frac{3 (x - 2)}{(x + 7) (x - 2)} - \frac{9}{(x - 2) (x + 7)} = 0$

Combine the numerators over the common denominator.

$\frac{x + 7 - 3 (x - 2)}{(x + 7) (x - 2)} - \frac{9}{(x - 2) (x + 7)} = 0$

Combine the numerators over the common denominator.

$\frac{x + 7 - 3 (x - 2) - 9}{(x + 7) (x - 2)} = 0$

Simplify each term.

Apply the distributive property.

$\frac{x + 7 - 3 x - 3 \cdot - 2 - 9}{(x + 7) (x - 2)} = 0$

Multiply $- 3$ by $- 2$ .

$\frac{x + 7 - 3 x + 6 - 9}{(x + 7) (x - 2)} = 0$

Subtract $3 x$ from $x$ .

$\frac{- 2 x + 7 + 6 - 9}{(x + 7) (x - 2)} = 0$

Add $7$ and $6$ .

$\frac{- 2 x + 13 - 9}{(x + 7) (x - 2)} = 0$

Subtract $9$ from $13$ .

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

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

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

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

Factor $2$ out of $4$ .

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

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $- 1$ .

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

Move the negative in front of the fraction.

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

Step 3

Set the denominator in $\frac{2}{x + 7}$ 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

Find Where Undefined/Discontinuous 1/(x-2)-3/(x+7)=9/(x^2+5x-14)

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