mun23
If a and b are each greater than x and y, which of the following must be true?
I. a + b > x + y
II. ab > xy
III. |a| + |b| > |x| + |y|
(A) I only
(B) II only
(C) I and II
(D) I and III
(E) I, II and III
Let's evaluate the given statements.
Statement I: a + b > x + y
Since a and b are each greater than x and y, a > x and b > y. Adding these two inequalities together, we obtain a + b > x + y. Statement I must be true.
Statement II: ab > xy
If a = b = -1 and x = y = -2, then the condition "a and b are each greater than x and y" is satisfied. However, ab = 1 is not greater than xy = 4. Statement II is not necessarily true.
Statement III: |a| + |b| > |x| + |y|
We can use the same numbers that we used in the previous statement. If a = b = -1 and x = y = -2, then |a| + |b| = 1 + 1 = 2, but |x| + |y| = 2 + 2 = 4. This shows that |a| + |b| is not necessarily greater than |x| + |y|. Statement III is not necessarily true.
Answer: A