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The problem asks to solve a system of inequalities:

\[
\begin{cases}
2x - 3 \leq 6 - x \\
4 - 2x > 6
\end{cases}
\]

### Step 1: Solve the first inequality

\[
2x - 3 \leq 6 - x
\]

- Add \(x\) to both sides:
\[
3x - 3 \leq 6
\]

- Add 3 to both sides:
\[
3x \leq 9
\]

- Divide by 3:
\[
x \leq 3
\]

### Step 2: Solve the second inequality

\[
4 - 2x > 6
\]

- Subtract 4 from both sides:
\[
-2x > 2
\]

- Divide by \(-2\) (remember to reverse the inequality):
\[
x < -1
\]

### Step 3: Combine the solutions

The solution from the first inequality is \(x \leq 3\), and from the second inequality, \(x < -1\). The intersection of these two solutions is:

\[
x < -1
\]

### Conclusion

The solution set is \(x < -1\). The set notation for this would be:

\[
(-\infty, -1)
\]

Would you like more details on any step, or do you have further questions?

Here are 5 related questions to deepen understanding:
1. How do you solve a system of inequalities with more than two inequalities?
2. What changes when dividing by a negative number in inequalities?
3. How do you graph the solution of an inequality on a number line?
4. What is the difference between strict and non-strict inequalities?
5. How would the solution change if one of the inequalities was an equality?

**Tip**: Always reverse the inequality sign when multiplying or dividing by a negative number in an inequality.

Resuelve e indique el conjunto solución en: {2x - 3 ≤ 6 - x, 4 - 2x > 6}

Learn how to solve the system of inequalities: {2x - 3 ≤ 6 - x, 4 - 2x > 6}. This problem covers key algebraic concepts, including solving linear inequalities and combining solutions, suitable for Grades 9-11.

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