Let's categorize the numbers into irrational numbers, integers, and rational numbers.

Given Set:

$\{-11, -6 \frac{1}{3}, -5, 0, 0.14, 3 \frac{5}{6}, 5, \sqrt{7}, 9\}$

1. Irrational Numbers:

Irrational numbers cannot be expressed as a fraction of two integers. They have non-terminating, non-repeating decimal expansions.

• $\sqrt{7}$

Irrational numbers: $\{\sqrt{7}\}$

2. Integers:

Integers are whole numbers, including positive numbers, negative numbers, and zero.

• $-11$
• $-5$
• $0$
• $5$
• $9$

Integers: $\{-11, -5, 0, 5, 9\}$

3. Rational Numbers:

Rational numbers can be expressed as a fraction of two integers. They have either terminating or repeating decimal expansions.

• $-11$ (integer)
• $-6 \frac{1}{3} = -\frac{19}{3}$
• $-5$ (integer)
• $0$ (integer)
• $0.14 = \frac{14}{100} = \frac{7}{50}$
• $3 \frac{5}{6} = \frac{23}{6}$
• $5$ (integer)
• $9$ (integer)

Rational numbers: $\{-11, -6 \frac{1}{3}, -5, 0, 0.14, 3 \frac{5}{6}, 5, 9\}$

Summary:

• Irrational numbers: $\{\sqrt{7}\}$
• Integers: $\{-11, -5, 0, 5, 9\}$
• Rational numbers: $\{-11, -6 \frac{1}{3}, -5, 0, 0.14, 3 \frac{5}{6}, 5, 9\}$

Would you like more details or have any questions?

Here are five related questions:

1. What are examples of irrational numbers other than square roots?
2. How do you determine if a number is rational or irrational?
3. Can a number be both rational and irrational? Why or why not?
4. Are all integers also rational numbers? Explain.
5. How do you express a repeating decimal as a fraction?

Tip: Remember that all integers are rational numbers, but not all rational numbers are integers!