To solve the inequality

$\frac{-2x^2 - 5x + 3}{x^2 - 4x - 5} \leq 0,$

we'll go through the following steps:

1. Factorize the numerator and the denominator.
2. Identify the values of $x$ where the expression is undefined or equals zero.
3. Set up test intervals to determine where the inequality holds.

Step 1: Factorize the Numerator and Denominator

1. Numerator: $-2x^2 - 5x + 3$

   • Factorizing $-2x^2 - 5x + 3$, we get: $-2x^2 - 5x + 3 = -(2x + 3)(x - 1).$

2. Denominator: $x^2 - 4x - 5$

   • Factorizing $x^2 - 4x - 5$, we get: $x^2 - 4x - 5 = (x - 5)(x + 1).$

The inequality now becomes: $\frac{-(2x + 3)(x - 1)}{(x - 5)(x + 1)} \leq 0.$

Step 2: Identify Critical Points

The critical points occur where the numerator or denominator is zero. These points are:

• From the numerator: $x = -\frac{3}{2}$ and $x = 1$.
• From the denominator: $x = 5$ and $x = -1$.

These points split the real line into intervals, which we will test to find where the inequality holds.

Step 3: Test Intervals

The intervals created by the critical points are:

1. $(-\infty, -1)$
2. $(-1, -\frac{3}{2})$
3. $(-\frac{3}{2}, 1)$
4. $(1, 5)$
5. $(5, \infty)$

We’ll choose test points in each interval and determine the sign of the expression in each interval.

1. Interval $(-\infty, -1)$: Choose $x = -2$.

   • The expression is positive in this interval.

2. Interval $(-1, -\frac{3}{2})$: Choose $x = -1.5$.

   • The expression is negative in this interval.

3. Interval $(-\frac{3}{2}, 1)$: Choose $x = 0$.

   • The expression is positive in this interval.

4. Interval $(1, 5)$: Choose $x = 2$.

   • The expression is negative in this interval.

5. Interval $(5, \infty)$: Choose $x = 6$.

   • The expression is positive in this interval.

Step 4: Determine Solution Intervals

We seek intervals where the expression is less than or equal to zero:

• The intervals where the expression is negative are $(-1, -\frac{3}{2})$ and $(1, 5)$.

Final Solution

The solution to the inequality is: $x \in \left( -1, -\frac{3}{2} \right] \cup \left[ 1, 5 \right).$

Would you like further details on any of the steps?

Here are some follow-up questions you might explore:

1. How can we verify the solution by substituting values back into the inequality?
2. What would the graph of this rational function look like?
3. How do we handle inequalities involving absolute values?
4. How do we interpret solutions that include both open and closed intervals?
5. What would happen if the inequality were strict (e.g., $< 0$) instead?

Tip: When solving rational inequalities, remember to exclude values where the denominator is zero from your solution, as these points make the expression undefined.