Let's solve the system of inequalities step by step:

### First inequality: 
\[
1 - x \geq -2
\]
1. Subtract 1 from both sides:
   \[
   -x \geq -3
   \]
2. Multiply both sides by -1 (remember to flip the inequality when multiplying or dividing by a negative number):
   \[
   x \leq 3
   \]
   
### Second inequality:
\[
2x - 3 > 1
\]
1. Add 3 to both sides:
   \[
   2x > 4
   \]
2. Divide both sides by 2:
   \[
   x > 2
   \]

### Combine the results:
- From the first inequality, \(x \leq 3\)
- From the second inequality, \(x > 2\)

Thus, the solution is:
\[
2 < x \leq 3
\]

This means that \(x\) is greater than 2 but less than or equal to 3.

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Would you like more details or have any questions? 

Here are 5 related questions you might want to explore:
1. How would the solution change if the second inequality was \(2x - 3 \geq 1\)?
2. What happens if we multiply both inequalities by a negative number at the start?
3. How do we graph the solution on a number line?
4. Can you solve the inequality if \(x\) is an integer?
5. How do compound inequalities work in more complex systems?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number!

This page explains how to solve the system of inequalities 1 − x ≥ −2 and 2x − 3 > 1. It includes step-by-step instructions on how to manipulate inequalities, reverse inequality signs when multiplying or dividing by negative numbers, and provides a detailed solution. Perfect for students in Grades 8-10 learning about algebra and linear inequalities.

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