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

Question

UpStudy AI · Accepted Answer

To solve the equation:

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

**Step 1: Combine the Terms**

Start by isolating the fractional term:

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

**Step 2: Eliminate the Denominator**

Multiply both sides of the equation by $$ x + 1 $$ to eliminate the denominator. **Note:** $$ x \neq -1 $$ since it would make the denominator zero.

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

$$
x^{2} - x - 2 = -x - 1
$$

**Step 3: Move All Terms to One Side**

Add $$ x + 1 $$ to both sides to bring all terms to the left-hand side of the equation:

$$
x^{2} - x - 2 + x + 1 = 0
$$

Simplify the equation:

$$
x^{2} - 1 = 0
$$

**Step 4: Solve for $$ x $$**

$$
x^{2} = 1
$$

Take the square root of both sides:

$$
x = \pm 1
$$

**Step 5: Check for Extraneous Solutions**

Recall that $$ x \neq -1 $$ (to avoid division by zero in the original equation). Therefore, $$ x = -1 $$ is not a valid solution.

**Final Solution:**

$$
x = 1
$$
$$ x = 1 $$

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