This image contains a variety of inequality problems, both quadratic and rational in nature. They are mostly grouped into sections with varying types of inequalities.

I will solve a few of them as examples, and you can ask for more if needed. Let's start with some quadratic inequalities.

Example 1:

Problem: $(x - 5.7)(x - 7.2) > 0$

1. Find the critical points (the values where the expression equals zero). These occur when:

   • $x - 5.7 = 0 \Rightarrow x = 5.7$
   • $x - 7.2 = 0 \Rightarrow x = 7.2$

2. Break the real number line into intervals based on these critical points: $(-\infty, 5.7), (5.7, 7.2), (7.2, \infty)$.

3. Test each interval by choosing values:

   • For $x = 5$ (from $(-\infty, 5.7)$): $(5 - 5.7)(5 - 7.2) = (-0.7)(-2.2) > 0$ → True.
   • For $x = 6$ (from $(5.7, 7.2)$): $(6 - 5.7)(6 - 7.2) = (0.3)(-1.2) < 0$ → False.
   • For $x = 8$ (from $(7.2, \infty)$): $(8 - 5.7)(8 - 7.2) = (2.3)(0.8) > 0$ → True.

4. Therefore, the solution is $(-\infty, 5.7) \cup (7.2, \infty)$.

Example 2:

Problem: $x^2 > x$

1. Rewrite the inequality: $x^2 - x > 0$.

2. Factor: $x(x - 1) > 0$.

3. Critical points are $x = 0$ and $x = 1$.

4. Test intervals:

   • For $x = -1$ (from $(-\infty, 0)$): $(-1)(-1 - 1) = (-1)(-2) > 0$ → True.
   • For $x = 0.5$ (from $(0, 1)$): $(0.5)(0.5 - 1) = (0.5)(-0.5) < 0$ → False.
   • For $x = 2$ (from $(1, \infty)$): $(2)(2 - 1) = (2)(1) > 0$ → True.

5. The solution is $(-\infty, 0) \cup (1, \infty)$.

Would you like to see more examples from this list or specific types of inequalities?

Relative questions:

1. How do you solve quadratic inequalities using the number line method?
2. Can inequalities be solved graphically, and if so, how?
3. What happens when a quadratic inequality has no real roots?
4. How do rational inequalities differ in solution from polynomial inequalities?
5. How can critical points of rational functions be interpreted in inequality solutions?

Tip:

When solving inequalities, always remember to carefully test the intervals defined by the critical points, as the inequality might change sign across these intervals.