Certainly! Let's solve the given inequalities step-by-step.
The problem involves two separate conditions for [tex]\( x \)[/tex]:
1. [tex]\( 2 < x \leq 5 \)[/tex]
2. [tex]\( x > 7 \)[/tex]
We need to find the values of [tex]\( x \)[/tex] that satisfy either of these two conditions.
### Condition 1: [tex]\( 2 < x \leq 5 \)[/tex]
This inequality tells us that [tex]\( x \)[/tex] is greater than 2 and less than or equal to 5.
- This can be written in interval notation as: [tex]\( (2, 5] \)[/tex]
### Condition 2: [tex]\( x > 7 \)[/tex]
This inequality tells us that [tex]\( x \)[/tex] is greater than 7.
- This can be written in interval notation as: [tex]\( (7, \infty) \)[/tex]
### Combining the Solutions
Since the problem states that [tex]\( x \)[/tex] can satisfy either [tex]\( 2 < x \leq 5 \)[/tex] or [tex]\( x > 7 \)[/tex], we take the union of the two intervals:
- First interval: [tex]\( (2, 5] \)[/tex]
- Second interval: [tex]\( (7, \infty) \)[/tex]
Combining these, we get the final solution:
- [tex]\( (2, 5] \)[/tex] union [tex]\( (7, \infty) \)[/tex]
In interval notation, the solution to the given inequalities [tex]\( 2 < x \leq 5 \)[/tex] or [tex]\( x > 7 \)[/tex] is [tex]\( (2, 5] \)[/tex] or [tex]\( (7, \infty) \)[/tex], which we can write as:
[tex]\[ (2, 5] \cup (7, \infty) \][/tex]
Thus, the values of [tex]\( x \)[/tex] that satisfy the given conditions are:
[tex]\[ x \in (2, 5] \, \text{or} \, x \in (7, \infty) \][/tex]
