To determine which of the provided symbols would replace the [tex]$\odot$[/tex] and create an inequality with no solutions, let's analyze each option alongside the given condition [tex]\(x \leq 3\)[/tex]:
1. Option a) [tex]\( x > 3 \)[/tex] and [tex]\( x \leq 3 \)[/tex]:
- Here, the condition [tex]\( x > 3 \)[/tex] implies that [tex]\(x\)[/tex] must be greater than 3.
- The condition [tex]\( x \leq 3 \)[/tex] implies that [tex]\(x\)[/tex] must be less than or equal to 3.
- No number can simultaneously be greater than 3 and less than or equal to 3.
- Therefore, there are no solutions for this combined inequality.
2. Option b) [tex]\( x < 3 \)[/tex] and [tex]\( x \leq 3 \)[/tex]:
- The condition [tex]\( x < 3 \)[/tex] implies [tex]\(x\)[/tex] must be less than 3.
- The condition [tex]\( x \leq 3 \)[/tex] implies [tex]\(x\)[/tex] must be less than or equal to 3.
- Any number that is less than 3 is also less than or equal to 3.
- Hence, this combined inequality has solutions.
3. Option c) [tex]\( x \geq 3 \)[/tex] and [tex]\( x \leq 3 \)[/tex]:
- The condition [tex]\( x \geq 3 \)[/tex] implies that [tex]\(x\)[/tex] must be greater than or equal to 3.
- The condition [tex]\( x \leq 3 \)[/tex] implies that [tex]\(x\)[/tex] must be less than or equal to 3.
- The only solution that satisfies both conditions is [tex]\( x = 3 \)[/tex].
- Hence, this combined inequality has solutions.
4. Option d) [tex]\( x \leq 3 \)[/tex] and [tex]\( x \leq 3 \)[/tex]:
- Both conditions are identical.
- The condition [tex]\( x \leq 3 \)[/tex] is repeated.
- Any number that satisfies [tex]\( x \leq 3 \)[/tex] in one part will satisfy it in the repeated part.
- Hence, this combined inequality has solutions.
From the above analysis, the symbol that replaces the [tex]$\odot$[/tex] and creates an inequality with no solutions is the ">" symbol.
Thus, the answer is: (">")
