Let's analyze each set of compound inequalities step-by-step to determine which one has no solution.
### Set 1:
1. [tex]\(2 x - 1 < x - 4\)[/tex]
2. [tex]\(-4 x + 3 > 2 x - 9\)[/tex]
Solution for Inequality 1:
[tex]\[
2 x - 1 < x - 4 \implies x < -3
\][/tex]
Solution for Inequality 2:
[tex]\[
-4 x + 3 > 2 x - 9 \implies -6 x > -12 \implies x < 2
\][/tex]
Combined Solution:
Combining [tex]\(x < -3\)[/tex] and [tex]\(x < 2\)[/tex] gives:
[tex]\[
x < -3
\][/tex]
### Set 2:
1. [tex]\(2 x - 5 \leq 3 x + 5\)[/tex]
2. [tex]\(-4 x - 1 > 2 x + 3\)[/tex]
Solution for Inequality 1:
[tex]\[
2 x - 5 \leq 3 x + 5 \implies -x \leq 10 \implies x \geq -10
\][/tex]
Solution for Inequality 2:
[tex]\[
-4 x - 1 > 2 x + 3 \implies -6 x > 4 \implies x < -\frac{2}{3}
\][/tex]
Combined Solution:
Combining [tex]\(x \geq -10\)[/tex] and [tex]\(x < -\frac{2}{3}\)[/tex] gives:
[tex]\[
-10 \leq x < -\frac{2}{3}
\][/tex]
### Set 3:
1. [tex]\(-3 x - 1 \leq 2 x + 9\)[/tex]
2. [tex]\(-4 x + 3 > -2 x + 11\)[/tex]
Solution for Inequality 1:
[tex]\[
-3 x - 1 \leq 2 x + 9 \implies -5 x \leq 10 \implies x \geq -2
\][/tex]
Solution for Inequality 2:
[tex]\[
-4 x + 3 > -2 x + 11 \implies -2 x > 8 \implies x < -4
\][/tex]
Combined Solution:
Combining [tex]\(x \geq -2\)[/tex] and [tex]\(x < -4\)[/tex] gives:
There is no overlap between [tex]\(x \geq -2\)[/tex] and [tex]\(x < -4\)[/tex], hence, this set has no solution.
### Set 4:
1. [tex]\(3(x + 1) \geq 2(x + 5)\)[/tex]
2. [tex]\(-2(x + 1) \leq 3(x + 1)\)[/tex]
Solution for Inequality 1:
[tex]\[
3(x + 1) \geq 2(x + 5) \implies 3x + 3 \geq 2x + 10 \implies x \geq 7
\][/tex]
Solution for Inequality 2:
[tex]\[
-2(x + 1) \leq 3(x + 1) \implies -2x - 2 \leq 3x + 3 \implies -5x \leq 5 \implies x \geq -1
\][/tex]
Combined Solution:
Combining [tex]\(x \geq 7\)[/tex] and [tex]\(x \geq -1\)[/tex] gives:
[tex]\[
x \geq 7
\][/tex]
### Conclusion:
The third set of compound inequalities has no solution, as there is no value of [tex]\(x\)[/tex] that satisfies both inequalities simultaneously.
So, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
