Let's break down the inequality to find the correct interval notation.

The inequality is:

$-3 < x \leq 3 \quad \text{or} \quad 7 \leq x$

First part: $-3 < x \leq 3$

This translates to the interval: $(-3, 3]$

• Parentheses for $-3$ because it is not included.
• Bracket for $3$ because it is included.

Second part: $7 \leq x$

This translates to the interval: $[7, \infty)$

• Bracket for $7$ because it is included.
• Parentheses for $\infty$ because $\infty$ is not a number and can't be included.

Now, combining both parts with the union symbol ($\cup$), the correct interval notation is: $(-3, 3] \cup [7, \infty)$

Conclusion:

The correct option is: $(-3, 3] \cup [7, \infty)$

Do you want more details or have any other questions? Here are some related problems:

1. What is the interval notation for $x > -2$ and $x \leq 4$?
2. How do we express $x < 0$ or $x \geq 5$ in interval notation?
3. Convert $1 \leq x < 10$ to interval notation.
4. Solve and express $x > 3$ and $x \leq 8$ in interval notation.
5. How would you graph the inequality $x \geq 2$ on a number line?

Tip: Always remember that parentheses $()$ mean the endpoint is excluded, while brackets $[]$ mean the endpoint is included!