To determine which one of the given compound inequalities has no solution, we need to solve each pair of inequalities step-by-step and analyze whether a common solution exists for each pair.
### First Compound Inequality
1. Inequality 1:
[tex]\[ 3x + 1 < 5x + 7 \][/tex]
[tex]\[ 1 - 7 < 5x - 3x \][/tex]
[tex]\[ -6 < 2x \][/tex]
[tex]\[ x > -3 \][/tex]
2. Inequality 2:
[tex]\[ -2x + 5 \leq -5x - 10 \][/tex]
[tex]\[ 5 + 10 \leq -5x + 2x \][/tex]
[tex]\[ 15 \leq -3x \][/tex]
[tex]\[ x \leq -5 \][/tex]
Combining these, we need [tex]\( x > -3 \)[/tex] and [tex]\( x \leq -5 \)[/tex].
#### Combined Solution:
There is no [tex]\( x \)[/tex] that satisfies [tex]\( x > -3 \)[/tex] and [tex]\( x \leq -5 \)[/tex] simultaneously. Thus, this compound inequality has no solution.
### Second Compound Inequality
1. Inequality 1:
[tex]\[ -2x + 9 > 4x + 3 \][/tex]
[tex]\[ 9 - 3 > 4x + 2x \][/tex]
[tex]\[ 6 > 6x \][/tex]
[tex]\[ x < 1 \][/tex]
2. Inequality 2:
[tex]\[ -8x - 9 < -7x - 3 \][/tex]
[tex]\[ -9 + 3 < -7x + 8x \][/tex]
[tex]\[ -6 < x \][/tex]
[tex]\[ x > -6 \][/tex]
Combining these, we need [tex]\( x < 1 \)[/tex] and [tex]\( x > -6 \)[/tex].
#### Combined Solution:
The combined solution is [tex]\( -6 < x < 1 \)[/tex], which is a valid range.
### Third Compound Inequality
1. Inequality 1:
[tex]\[ -2(x - 9) < 3(x + 4) \][/tex]
[tex]\[ -2x + 18 < 3x + 12 \][/tex]
[tex]\[ 18 - 12 < 3x + 2x \][/tex]
[tex]\[ 6 < 5x \][/tex]
[tex]\[ x > \frac{6}{5} \][/tex]
2. Inequality 2:
[tex]\[ -4(x - 1) > -5(x - 2) \][/tex]
[tex]\[ -4x + 4 > -5x + 10 \][/tex]
[tex]\[ 4 - 10 > -5x + 4x \][/tex]
[tex]\[ -6 > -x \][/tex]
[tex]\[ x > 6 \][/tex]
Combining these, we need [tex]\( x > \frac{6}{5} \)[/tex] and [tex]\( x > 6 \)[/tex].
#### Combined Solution:
The combined solution is [tex]\( x > 6 \)[/tex], which is a valid range.
### Fourth Compound Inequality
1. Inequality 1:
[tex]\[ 3x - 7 \leq 5x + 5 \][/tex]
[tex]\[ -7 - 5 \leq 5x - 3x \][/tex]
[tex]\[ -12 \leq 2x \][/tex]
[tex]\[ x \geq -6 \][/tex]
2. Inequality 2:
[tex]\[ -4x + 8 < 5x - 1 \][/tex]
[tex]\[ 8 + 1 < 5x + 4x \][/tex]
[tex]\[ 9 < 9x \][/tex]
[tex]\[ x > 1 \][/tex]
Combining these, we need [tex]\( x \geq -6 \)[/tex] and [tex]\( x > 1 \)[/tex].
#### Combined Solution:
The combined solution is [tex]\( x > 1 \)[/tex], which is a valid range.
### Conclusion
The first compound inequality has no solution.
[tex]\[
\boxed{3x + 1 < 5x + 7 \text{ and } -2x + 5 \leq -5x - 10}
\][/tex]
