Question

.

Answer 1

@ikawasalmon

Answer 2

To solve the rational inequality \(\frac{x - 3}{x + 1} \geq 2\), we first need to rewrite the inequality in a standard form. Start by subtracting 2 from both sides:

\[
\frac{x - 3}{x + 1} - 2 \geq 0
\]

Next, we need a common denominator to combine the fractions. The common denominator will be \(x + 1\):

\[
\frac{x - 3}{x + 1} - \frac{2(x + 1)}{x + 1} \geq 0
\]

This simplifies to:

\[
\frac{x - 3 - 2(x + 1)}{x + 1} \geq 0
\]

Now, distribute the \(-2\) in the numerator:

\[
\frac{x - 3 - 2x - 2}{x + 1} \geq 0
\]

Combining like terms gives:

\[
\frac{-x - 5}{x + 1} \geq 0
\]

Now we can change the sign of the inequality by multiplying both sides by \(-1\) (remembering that we need to flip the inequality sign):

\[
\frac{x + 5}{x + 1} \leq 0
\]

Next, we will find the points where the fraction is equal to zero or undefined.

1. **Zeroes of the numerator**: Set \(x + 5 = 0\):
\[
x = -5
\]

2. **Undefined points of the denominator**: Set \(x + 1 = 0\):
\[
x = -1
\]

Now we have two critical points, \(x = -5\) and \(x = -1\). We will test intervals determined by these points in the inequality \(\frac{x + 5}{x + 1} \leq 0\). The intervals are:

- \( (-\infty, -5) \)
- \( (-5, -1) \)
- \( (-1, \infty) \)

### Testing Intervals

1. **Interval \( (-\infty, -5) \)**: Choose \(x = -6\):
\[
\frac{-6 + 5}{-6 + 1} = \frac{-1}{-5} = \frac{1}{5} > 0 \quad \text{(not a solution)}
\]

2. **Interval \( (-5, -1) \)**: Choose \(x = -3\):
\[
\frac{-3 + 5}{-3 + 1} = \frac{2}{-2} = -1 < 0 \quad \text{(solution)}
\]

3. **Interval \( (-1, \infty) \)**: Choose \(x = 0\):
\[
\frac{0 + 5}{0 + 1} = \frac{5}{1} = 5 > 0 \quad \text{(not a solution)}
\]

### Conclusion

The inequality \(\frac{x + 5}{x + 1} \leq 0\) holds in the interval \((-5, -1)\).

At the critical points:

- At \(x = -5\), \(\frac{-5 + 5}{-5 + 1} = \frac{0}{-4} = 0\) (included).
- At \(x = -1\), the expression is undefined (not included).

Thus, the solution to the inequality \(\frac{x - 3}{x + 1} \geq 2\) is:

\[
\boxed{[-5, -1)}
\]