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

3. In a class of 120 students, 70 offer Mathematics, 65 offer Physics, and 30 offer both. How many students offer neither subject? Step 1: Find the number of students who offer at least one 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 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: |M P| = |M| + |P| - |M P| Substitute the given values: |M P| = 70 + 65 - 30 |M P| = 135 - 30 |M P| = 105 So, 105 students offer at least one subject. Step 2: Find the number of students who offer neither 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. Neither = Total students - |M P| Neither = 120 - 105 Neither = 15 The correct option is A. A. 15 4. Solve the inequality: 3 - 2x 5 Step 1: Subtract 3 from both sides of the inequality. 3 - 2x 5 -2x 5 - 3 -2x 2 Step 2: Divide both sides by -2. Remember to reverse the inequality sign when dividing or multiplying by a negative number. (-2x)/(-2) (2)/(-2) x -1 The correct option is A. A. x -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 (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. (1)/(4) (1)/(4) = 0.25 This is a terminating decimal. (Denominator 4 = 2^2) B. (1)/(5) (1)/(5) = 0.2 This is a terminating decimal. (Denominator 5 = 5^1) C. (1)/(3) (1)/(3) = 0.333... This is a non-terminating recurring decimal. (Denominator 3 has a prime factor other than 2 or 5) D. (1)/(8) (1)/(8) = 0.125 This is a terminating decimal. (Denominator 8 = 2^3) The correct option is C. C. (1)/(3) 6. If f(x) = sqrt(x - 2), find the domain of f. Step 1: Identify the restriction for the function. 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. Step 2: Set up the inequality. In this function, u = x - 2. Therefore, we must have: x - 2 0 Step 3: Solve the inequality for x. Add 2 to both sides: x 2 The domain of f(x) is all real numbers x such that x 2. The correct option is A. A. x 2