In a class of 120 students, 70 offer Mathematics, 65 offer Physics, and 30 offer both. How many students offer neither subject?

Let $M$ be the set of students who offer Mathematics and $P$ be the set of students who offer Physics.
We are given:

• Total students = 120
• $|M| = 70$
• $|P| = 65$
• $|M \cap P| = 30$ (students who offer both)

The number of students who offer Mathematics or Physics (or both) is given by the formula for the union of two sets:

Substitute the given values:

So, 105 students offer at least one subject.

The number of students who offer neither subject is the total number of students minus the number of students who offer at least one subject.

The correct option is A.
$\boxed{\text{A. } 15}$

4. Solve the inequality: $3 - 2x \ge 5$

Remember to reverse the inequality sign when dividing or multiplying by a negative number.

The correct option is A.
$\boxed{\text{A. } x \le -1}$

5. Which of the following is a non-terminating recurring decimal?

A non-terminating recurring decimal is a decimal that continues infinitely with a repeating pattern of digits. A fraction $\frac{p}{q}$ (in simplest form) results in a terminating decimal if the prime factors of the denominator $q$ are only 2s and/or 5s. Otherwise, it results in a non-terminating recurring decimal.

Let's convert each option to a decimal:

• A. $\frac{1}{4}$

  $$\frac{1}{4} = 0.25$$

  This is a terminating decimal. (Denominator $4 = 2^2$)

• B. $\frac{1}{5}$

  $$\frac{1}{5} = 0.2$$

  This is a terminating decimal. (Denominator $5 = 5^1$)

• C. $\frac{1}{3}$

  $$\frac{1}{3} = 0.333...$$

  This is a non-terminating recurring decimal. (Denominator 3 has a prime factor other than 2 or 5)

• D. $\frac{1}{8}$

  $$\frac{1}{8} = 0.125$$

  This is a terminating decimal. (Denominator $8 = 2^3$)

The correct option is C.
$\boxed{\text{C. } \frac{1}{3}}$

6. If $f(x) = \sqrt{x - 2}$, find the domain of $f$.

For a square root function $f(x) = \sqrt{u}$, the expression under the square root, $u$, must be non-negative (greater than or equal to zero) for the function to produce real numbers.

In this function, $u = x - 2$. Therefore, we must have:

Add 2 to both sides:

The domain of $f(x)$ is all real numbers $x$ such that $x \ge 2$.

The correct option is A.
$\boxed{\text{A. } x \ge 2}$