Find Where Undefined/Discontinuous x/(x+1)-x/(x^2+3x+2)=4/(x+2)

$\frac{x}{x + 1} - \frac{x}{x^{2} + 3 x + 2} = \frac{4}{x + 2}$

Step 1

Subtract $\frac{4}{x + 2}$ from both sides of the equation.

$\frac{x}{x + 1} - \frac{x}{x^{2} + 3 x + 2} - \frac{4}{x + 2} = 0$

Step 2

Simplify $\frac{x}{x + 1} - \frac{x}{x^{2} + 3 x + 2} - \frac{4}{x + 2}$ .

Factor $x^{2} + 3 x + 2$ 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 $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Simplify the numerator.

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

Factor $x$ out of $- x$ .

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

Factor $x$ out of $x (x + 2) + x \cdot - 1$ .

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

Subtract $1$ from $2$ .

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

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Step 3

Set the denominator in $\frac{x - 4}{x + 2}$ equal to $0$ to find where the expression is undefined.

$x + 2 = 0$

Step 4

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

Find Where Undefined/Discontinuous x/(x+1)-x/(x^2+3x+2)=4/(x+2)

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