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Subtract \(\frac{x}{n}\) from both sides to get: \(\frac{y}{n}<\frac{z}{n}\)
There are two possible cases to consider: n is POSITIVE and n is NEGATIVE
If n is POSITIVE, we can multiply both sides by n to get: \(y < z\) If n is NEGATIVE, we can multiply both sides by n to get: \(y > z\)
REPHRASED target question:Is y < z?
Aside: the video below has tips on rephrasing the target question
Statement 1: \(x^2-z+y<0\) Add z to both sides to get: \(x^2+y<z\) Subtract y from both sides to get: \(x^2<z-y\) Since 0 ≤ x², we can write 0 ≤ x² < z - y This tells us that: 0 < z - y Add y to both sides to get: y < z So, the answer to the REPHRASED target question is YES, y IS less than z Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT
Statement 2: \(xy<xz\) There are several values of x, y, z and n that satisfy statement 2 (and the given information). Here are two: Case a: x = 1, y = 2, z = 3 and n = 1. In this case, the answer to the REPHRASED target question is YES, y IS less than z Case b: x = 1, y = 3, z = 2 and n = -1. In this case, the answer to the REPHRASED target question is NO, y is NOT less than z Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT