If $X < Z$ and $X < Y$, which of the following is necessarily true? I. $Y < Z$ II. $X^2 < YZ$ III. $ZX < Y + Z$

### Concept & Strategy When asked what is "necessarily true" in inequalities, finding just one valid counterexample for a statement is enough to prove it false. Real numbers include positive integers, negative integers, and fractions. ### Step-by-Step Solution 1. We are given $X < Z$ and $X < Y$. This only tells us $X$ is the smallest among the three variables. It defines no relationship between $Y$ and $Z$, nor does it restrict their signs (positive/negative). 2. Evaluate I ($Y < Z$): Since we only know $X$ is less than both, $Y$ could be greater than, less than, or equal to $Z$. Not necessarily true. 3. Evaluate II ($X^2 < YZ$): Test with negative numbers. Let $X = -2, Y = -1, Z = 0$. This satisfies $X < Y$ and $X < Z$. Substituting these into II gives $(-2)^2 < (-1)(0) \Rightarrow 4 < 0$, which is false. Not necessarily true. 4. Evaluate III ($ZX < Y + Z$): Using the same values $X = -2, Y = -1, Z = 0$, substitute into III. $(0)(-2) < (-1) + 0 \Rightarrow 0 < -1$, which is false. Not necessarily true. 5. Since all three statements fail under specific conditions, none of them are necessarily true. ### Exam Strategy & Shortcut The fastest way to test generic variable inequalities is to use negative integers or zero, as they frequently flip signs or collapse terms, exposing logical flaws that positive numbers might hide. Always try one set of negative values as a baseline test. ### Common Pitfall Assuming all variables are positive natural numbers. If you substitute $X=1, Y=2, Z=3$, statements II and III might appear to be correct, leading to a wrong choice. Always test edge cases (negatives, zero, fractions). ### Final Answer **Therefore, the correct answer is None of these.**