MPT Case Analysis 3 — Questions and Answers
Question 1: Analyze by cases: for the equation |x - 3| = 2x, which values of x are valid solutions?
- x = 3 only
- x = 1 only (Correct answer)
- x = 1 and x = 3
- No solution
Correct answer: x = 1 only
Case 1: x - 3 = 2x gives x = -3, but 2(-3) = -6 < 0 so invalid; Case 2: -(x-3) = 2x gives x = 1, check: |1-3| = 2 = 2(1) ✓.
Question 2: A piecewise function f(x) = -x if x < 0, f(x) = 0 if x = 0, f(x) = x if x > 0. This function is best described as:
- A linear function
- The absolute value function |x| (Correct answer)
- A step function
- A quadratic function
Correct answer: The absolute value function |x|
The definition matches |x| exactly: negative inputs become positive, zero stays zero, positive inputs remain unchanged.
Question 3: Solve by case analysis: if x² > 9, what is the solution set?
- x > 3
- x < -3 or x > 3 (Correct answer)
- -3 < x < 3
- x > -3
Correct answer: x < -3 or x > 3
x² > 9 means |x| > 3, which splits into x > 3 or x < -3.
Question 4: A store charges $5 per item for 1–10 items and $4 per item for more than 10 items. Using case analysis, what is the cost for exactly 10 items?
- $40
- $45
- $50 (Correct answer)
- $44
Correct answer: $50
Exactly 10 items falls in the first pricing tier (1–10 items), so cost = 10 × $5 = $50.
Question 5: For f(x) = |x - 2| + |x + 1|, what is the minimum value of f(x)?
- 1
- 2
- 3 (Correct answer)
- 0
Correct answer: 3
For -1 ≤ x ≤ 2, f(x) = (2 - x) + (x + 1) = 3 (constant); outside this interval f(x) > 3, so the minimum is 3.
Question 6: Case analysis on a quadratic: when does x² - 5x + 6 > 0?
- x < 2 or x > 3 (Correct answer)
- 2 < x < 3
- x < -2 or x > -3
- Always
Correct answer: x < 2 or x > 3
Factor: (x - 2)(x - 3) > 0; both factors positive gives x > 3, both negative gives x < 2.
Question 7: A function is defined as f(x) = 2 if x is rational, and f(x) = 0 if x is irrational. What is f(π) + f(1/2)?
- 2 (Correct answer)
- 4
- 0
- 1
Correct answer: 2
π is irrational so f(π) = 0; 1/2 is rational so f(1/2) = 2; sum = 0 + 2 = 2.
MPT Case Analysis 3
Analyze by cases: for the equation |x - 3| = 2x, which values of x are valid solutions?