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I see that you've uploaded an image containing three inequalities in need of solutions. Let's go step by step and solve them.

### Problem 1:
\[
\frac{x}{2} \geq \frac{1 - x}{4 - 2x}
\]
We will solve this inequality by eliminating the fractions and simplifying.

### Problem 2:
\[
\frac{x - 2}{x^2} \leq \frac{x + 1}{x - 3}
\]
This inequality involves rational functions, so we need to analyze the critical points where the numerator or denominator is zero, then determine the sign of the expressions in different intervals.

### Problem 3:
\[
12 - x^1 + 3 - 2x^1 \leq 3
\]
It looks like this is a linear inequality, and we need to simplify and solve it.

Let me go ahead and solve each one for you!Here are the solutions for the three inequalities:

### Problem 1:
\[
\frac{x}{2} \geq \frac{1 - x}{4 - 2x}
\]
The critical values for \( x \) are:
\[
x = \frac{3}{2} - \frac{\sqrt{5}}{2} \quad \text{and} \quad x = \frac{3}{2} + \frac{\sqrt{5}}{2}
\]
These are approximate solutions, and the inequality will need further analysis to determine the intervals for \( x \).

### Problem 2:
\[
\frac{x - 2}{x^2} \leq \frac{x + 1}{x - 3}
\]
The critical points are \( x = 1 \), but the other roots are complex:
\[
x = -\frac{1}{2} \pm \frac{\sqrt{23}}{2}i
\]
This means the inequality may not have real solutions other than at \( x = 1 \), but further analysis of the intervals and sign changes should be done.

### Problem 3:
\[
12 - x + 3 - 2x \leq 3
\]
The solution is:
\[
x = 4
\]

Would you like more detailed steps or an explanation for any of these problems?

Here are some related questions:
1. How can we analyze rational inequalities using a sign chart?
2. What is the importance of checking for domain restrictions in inequalities?
3. How do you handle complex roots in solving inequalities?
4. Can rational functions have asymptotes affecting inequality solutions?
5. How do we simplify inequalities that involve linear terms?

**Tip**: When solving rational inequalities, always check for critical points by setting both the numerator and denominator equal to zero.

Explore solutions to complex inequalities involving rational functions and linear terms. Learn how to solve inequalities step-by-step with critical point analysis and interval evaluation. Suitable for advanced high school students and beyond.

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