To determine which of the given compound inequalities has no solution, we need to analyze each one individually.
1. [tex]\( x \leq 2 \)[/tex] and [tex]\( x > 7 \)[/tex]:
- This is a conjunction (an "and" statement).
- [tex]\( x \leq 2 \)[/tex] means that [tex]\( x \)[/tex] can be at most 2.
- [tex]\( x > 7 \)[/tex] means that [tex]\( x \)[/tex] must be greater than 7.
- There is no number that can simultaneously be at most 2 and greater than 7.
- Therefore, this compound inequality has no solution.
2. [tex]\( x \leq 4 \)[/tex] and [tex]\( x > 1 \)[/tex]:
- This is a conjunction (an "and" statement).
- [tex]\( x \leq 4 \)[/tex] means that [tex]\( x \)[/tex] can be at most 4.
- [tex]\( x > 1 \)[/tex] means that [tex]\( x \)[/tex] must be greater than 1.
- There are numbers that satisfy both conditions. For example, [tex]\( x = 2 \)[/tex] is a solution since it is less than or equal to 4 and greater than 1.
- Therefore, this compound inequality has a solution.
3. [tex]\( x \leq 2 \)[/tex] or [tex]\( x > 7 \)[/tex]:
- This is a disjunction (an "or" statement).
- [tex]\( x \leq 2 \)[/tex] means that [tex]\( x \)[/tex] can be at most 2.
- [tex]\( x > 7 \)[/tex] means that [tex]\( x \)[/tex] must be greater than 7.
- Since this is an "or" statement, it is sufficient for [tex]\( x \)[/tex] to meet either one of the conditions.
- There are numbers that satisfy [tex]\( x \leq 2 \)[/tex], such as [tex]\( x = 1 \)[/tex].
- There are also numbers that satisfy [tex]\( x > 7 \)[/tex], such as [tex]\( x = 8 \)[/tex].
- Therefore, this compound inequality has a solution.
4. [tex]\( 2 \leq x < 7 \)[/tex]:
- This is a compound inequality with a single variable.
- [tex]\( 2 \leq x \)[/tex] means that [tex]\( x \)[/tex] must be at least 2.
- [tex]\( x < 7 \)[/tex] means that [tex]\( x \)[/tex] must be less than 7.
- There are numbers that satisfy both conditions. For example, [tex]\( x = 3 \)[/tex] is a solution since it is greater than or equal to 2 and less than 7.
- Therefore, this compound inequality has a solution.
Given the analysis above, the compound inequality for which there is no solution is:
[tex]\[ \boxed{x \leq 2 \text{ and } x > 7} \][/tex]
