Solve by Factoring 2/(x+1)-1/(x-2)=-1/2

$\frac{2}{x + 1} - \frac{1}{x - 2} = - \frac{1}{2}$

Step 1

Add $\frac{1}{2}$ to both sides of the equation.

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

Step 2

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $(x - 2) (x + 1)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $- 2$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $x$ from $2 x$ .

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

Subtract $1$ from $- 4$ .

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

To write $\frac{x - 5}{(x + 1) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

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

Multiply $\frac{x - 5}{(x + 1) (x - 2)}$ by $\frac{2}{2}$ .

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

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

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

Reorder the factors of $(x + 1) (x - 2) \cdot 2$ .

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

Reorder the factors of $2 ((x + 1) (x - 2))$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Move $2$ to the left of $x$ .

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

Multiply $- 5$ by $2$ .

$\frac{2 \cdot x - 10 + (x + 1) (x - 2)}{2 (x + 1) (x - 2)} = 0$

Expand $(x + 1) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 x - 10 + x (x - 2) + 1 (x - 2)}{2 (x + 1) (x - 2)} = 0$

Apply the distributive property.

$\frac{2 x - 10 + x \cdot x + x \cdot - 2 + 1 (x - 2)}{2 (x + 1) (x - 2)} = 0$

Apply the distributive property.

$\frac{2 x - 10 + x \cdot x + x \cdot - 2 + 1 x + 1 \cdot - 2}{2 (x + 1) (x - 2)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{2 x - 10 + x^{2} + x \cdot - 2 + 1 x + 1 \cdot - 2}{2 (x + 1) (x - 2)} = 0$

Move $- 2$ to the left of $x$ .

$\frac{2 x - 10 + x^{2} - 2 \cdot x + 1 x + 1 \cdot - 2}{2 (x + 1) (x - 2)} = 0$

Multiply $x$ by $1$ .

$\frac{2 x - 10 + x^{2} - 2 x + x + 1 \cdot - 2}{2 (x + 1) (x - 2)} = 0$

Multiply $- 2$ by $1$ .

$\frac{2 x - 10 + x^{2} - 2 x + x - 2}{2 (x + 1) (x - 2)} = 0$

Add $- 2 x$ and $x$ .

$\frac{2 x - 10 + x^{2} - x - 2}{2 (x + 1) (x - 2)} = 0$

Subtract $x$ from $2 x$ .

$\frac{x - 10 + x^{2} - 2}{2 (x + 1) (x - 2)} = 0$

Subtract $2$ from $- 10$ .

$\frac{x + x^{2} - 12}{2 (x + 1) (x - 2)} = 0$

Reorder terms.

$\frac{x^{2} + x - 12}{2 (x + 1) (x - 2)} = 0$

Factor $x^{2} + x - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

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

Step 3

Set the numerator equal to zero.

$(x - 3) (x + 4) = 0$

Step 4

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x + 4 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(x - 3) (x + 4) = 0$ true.

$x = 3, - 4$

Solve by Factoring 2/(x+1)-1/(x-2)=-1/2

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