Sure, let's analyze each compound inequality step by step to determine which one has no solution.
### Compound Inequality 1:
[tex]\( m + 7 < 8 \)[/tex] and [tex]\( -5m \geq 10 \)[/tex]
1. Solve [tex]\( m + 7 < 8 \)[/tex]:
[tex]\[
m + 7 < 8 \implies m < 1
\][/tex]
2. Solve [tex]\( -5m \geq 10 \)[/tex]:
[tex]\[
-5m \geq 10 \implies m \leq -2
\][/tex]
Combining these solutions, we need [tex]\( m < 1 \)[/tex] and [tex]\( m \leq -2 \)[/tex]. These conditions are incompatible since there is no number that is both less than 1 and less than or equal to -2. Therefore, this compound inequality has no solution.
### Compound Inequality 2:
[tex]\( 3m \leq -21 \)[/tex] and [tex]\( m + 24 > 19 \)[/tex]
1. Solve [tex]\( 3m \leq -21 \)[/tex]:
[tex]\[
3m \leq -21 \implies m \leq -7
\][/tex]
2. Solve [tex]\( m + 24 > 19 \)[/tex]:
[tex]\[
m + 24 > 19 \implies m > -5
\][/tex]
Combining these solutions, we need [tex]\( m \leq -7 \)[/tex] and [tex]\( m > -5 \)[/tex]. These conditions are incompatible since there is no number that is both less than or equal to -7 and greater than -5. Therefore, this compound inequality has no solution.
### Compound Inequality 3:
[tex]\( -3m < 6 \)[/tex] and [tex]\( 4m > 24 \)[/tex]
1. Solve [tex]\( -3m < 6 \)[/tex]:
[tex]\[
-3m < 6 \implies m > -2
\][/tex]
2. Solve [tex]\( 4m > 24 \)[/tex]:
[tex]\[
4m > 24 \implies m > 6
\][/tex]
Combining these solutions, we need [tex]\( m > -2 \)[/tex] and [tex]\( m > 6 \)[/tex]. Since [tex]\( m > 6 \)[/tex] implies that [tex]\( m > -2 \)[/tex] is already satisfied, the combined solution is [tex]\( m > 6 \)[/tex]. Therefore, this compound inequality does have a solution.
### Compound Inequality 4:
[tex]\( 3m + 7 \geq 7 \)[/tex] and [tex]\( -4m + 8 \leq -12 \)[/tex]
1. Solve [tex]\( 3m + 7 \geq 7 \)[/tex]:
[tex]\[
3m + 7 \geq 7 \implies 3m \geq 0 \implies m \geq 0
\][/tex]
2. Solve [tex]\( -4m + 8 \leq -12 \)[/tex]:
[tex]\[
-4m + 8 \leq -12 \implies -4m \leq -20 \implies m \geq 5
\][/tex]
Combining these solutions, we need [tex]\( m \geq 0 \)[/tex] and [tex]\( m \geq 5 \)[/tex]. The combined solution is [tex]\( m \geq 5 \)[/tex]. Therefore, this compound inequality does have a solution.
### Conclusion:
Both the first and second compound inequalities have no solution. Analyzing both, they both have incompatible conditions. Thus:
1. [tex]\( m + 7 < 8 \)[/tex] and [tex]\( -5m \geq 10 \)[/tex]
2. [tex]\( 3m \leq -21 \)[/tex] and [tex]\( m + 24 > 19 \)[/tex]
These compound inequalities have no solutions.
