The image you uploaded contains several mathematical inequalities and expressions. Let's go through each part step by step.

Part 1: Solving Inequalities

The inequalities in the first part of the image are:

1. $\frac{x - 2}{x^2 - 1} \leq 0$
2. $\frac{x - 1}{x^2 - 4} \leq 0$
3. $\frac{9 - x^2}{x^2 - 1} \geq 0$

Step 1: Find the critical points

For each inequality, find the points where the numerator and denominator equal zero, since these points can be potential solutions or create discontinuities in the inequalities.

1. Inequality 1: $\frac{x - 2}{x^2 - 1} \leq 0$

   • Numerator: $x - 2 = 0 \Rightarrow x = 2$
   • Denominator: $x^2 - 1 = 0 \Rightarrow (x - 1)(x + 1) = 0 \Rightarrow x = \pm 1$

2. Inequality 2: $\frac{x - 1}{x^2 - 4} \leq 0$

   • Numerator: $x - 1 = 0 \Rightarrow x = 1$
   • Denominator: $x^2 - 4 = 0 \Rightarrow (x - 2)(x + 2) = 0 \Rightarrow x = \pm 2$

3. Inequality 3: $\frac{9 - x^2}{x^2 - 1} \geq 0$

   • Numerator: $9 - x^2 = 0 \Rightarrow (3 - x)(3 + x) = 0 \Rightarrow x = \pm 3$
   • Denominator: $x^2 - 1 = 0 \Rightarrow x = \pm 1$

Step 2: Analyze the sign changes

The critical points split the number line into intervals. The sign of each expression within each interval needs to be checked.

Part 2: Rational Inequalities

This section contains complex rational inequalities. For these, each inequality should be individually simplified before solving. Here's an overview:

1. Expression 1: $\frac{3(6 - x^2) - 14x}{x^2 - 5x + 6} \leq \frac{3x - 10}{x - 3} + \frac{1}{x - 2}$

   Simplify the left-hand side (LHS) and right-hand side (RHS) separately, find a common denominator, and solve for $x$.

2. Expression 2: $\frac{10 - 3x}{x - 3} - \frac{15}{x - 2} \geq \frac{3(x^2 + 1) + 21}{x^2 - 5x + 6}$

   Similar steps as before: simplify both sides and solve for $x$.

3. Expression 3: $\frac{x - 5}{x^2 - 5x + 6} \leq \frac{4}{x - 3} + \frac{1 + x}{x - 2}$

   Again, simplify both sides, combine terms, and solve for $x$.

Next Steps:

Each inequality should be solved individually by:

1. Finding the critical points.
2. Testing the sign of the expression within each interval.
3. Writing down the solution sets for each inequality.

Would you like a detailed step-by-step solution for each inequality?

Related Questions:

1. How do you find the critical points of a rational function?
2. What are the steps for solving a quadratic inequality?
3. How do sign charts help in solving inequalities?
4. What are the common mistakes to avoid when solving rational inequalities?
5. Can you explain the process of finding the union and intersection of solution sets?

Tip:

Always check the domain of the function first, especially for rational expressions, as it will help avoid invalid solutions.