The question in the image asks to represent the set of real numbers greater than 5 in set-builder notation and interval notation.

Explanation:

1. Set-Builder Notation: A mathematical way to describe a set using a property that its elements must satisfy.

   • For numbers greater than 5, it is written as: $\{ x \mid x > 5 \}$ This reads: "The set of all $x$ such that $x > 5$."

2. Interval Notation: A way to represent subsets of real numbers on a number line using brackets and parentheses.

   • Since the set includes numbers greater than 5 (but not 5 itself), it is written as: $(5, \infty)$ Here:
     • Parentheses $()$ indicate that 5 is not included.
     • $\infty$ represents infinity, which is always open (not included).

Would you like me to explain further or provide a visualization? Here are some follow-up questions you might explore:

Follow-Up Questions:

1. How would you write the set of real numbers less than or equal to 7 in set-builder and interval notation?
2. How does the use of square brackets $[ ]$ differ from parentheses $( )$ in interval notation?
3. What does $(-\infty, 5)$ represent in interval notation?
4. Can a set be written using both notations at the same time? Why or why not?
5. How would you combine two intervals, such as $(5, 10)$ and $(15, \infty)$, using union notation?

Tip:

When working with set-builder and interval notations, always check whether the boundary points are included (closed) or excluded (open).