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I see you've uploaded an image containing six mathematical inequalities. Let's go over them step by step to solve each one. The equations are as follows:

1. \(\frac{x + 2}{4 - 2x} \geq 1 - x\)
2. \(\frac{x - 2}{x^2} \leq \frac{x + 1}{x}\)
3. \(|2 - x| + |3 - 2x| \leq 3\)
4. \(|x + 1|^2 + 2|x + 2| \geq 2\)
5. \(2x + 3 \geq |4x + 5|\)
6. \(|x| + 3x \leq 2\)

Let's start by solving the first inequality:

### 1. \(\frac{x + 2}{4 - 2x} \geq 1 - x\)

We'll proceed with the following steps:
- Combine terms on one side.
- Eliminate the denominator by multiplying through.
- Solve for \(x\).

I'll work through this now.The solution to the inequality \(\frac{x + 2}{4 - 2x} \geq 1 - x\) is:

\[
\left(\frac{7 - \sqrt{33}}{4} \leq x < 2 \right) \cup \left( \frac{7 + \sqrt{33}}{4} \leq x \right)
\]

This means:
- \(x\) lies between \(\frac{7 - \sqrt{33}}{4}\) and 2, or
- \(x \geq \frac{7 + \sqrt{33}}{4}\).

Would you like me to move on to solving the second inequality? 

Here are five related questions you might consider:
1. What happens if you change the inequality to a strict inequality (i.e., \(>\))?
2. How would this inequality change if you multiplied through by \(4 - 2x\)?
3. What are the critical points where the expression changes sign?
4. Can we solve this inequality graphically? 
5. How would you check the solution using substitution?

**Tip:** Always check for restrictions on variables (like denominators that cannot be zero) when solving inequalities.

Solve the following inequalities:
1. (x + 2) / (4 - 2x) ≥ 1 - x
2. (x - 2) / x^2 ≤ (x + 1) / x
3. |2 - x| + |3 - 2x| ≤ 3
4. |x + 1|^2 + 2|x + 2| ≥ 2
5. 2x + 3 ≥ |4x + 5|
6. |x| + 3x ≤ 2

Learn how to solve inequalities involving rational expressions, absolute values, and quadratic expansions. This problem set covers concepts such as rational inequalities, absolute value properties, and algebraic manipulations. Ideal for students in Grades 9-12.

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