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Substitution (Special Cases)

We already studied this method in the section Substitution. Here, we will practice more and learn a couple special cases.
Strategy: In a problem with two variables, say, x and y, you must check the case in which

x = y. (This often gives a double case.)

x and y are positive.
Column A Column B
Average of x and y Average of  x3 and y3

Let x = y = 1. Then Column A becomes . And Column B becomes . In this case, the columns are equal. But if x = y = 2, then Column A becomes  and Column B becomes . In this case, the columns are unequal. This is a double case and therefore the answer is (D).


x and y are integers greater than or equal to 1.
Column A Column B
2x + y 2x + 2y

If x ≠ y, then Column A is larger than Column B. (Plug in a few numbers until you are convinced.) But if x = y = 1, then the columns are equal: 2x + y = 21 + 1 = 22 = 4 and 2x + 2y = 21 + 21 = 4. Hence, there is not enough information to decide, and the answer is (D).


Strategy: When you are given x < 0, you must plug in negative whole numbers, negative fractions, and –1. (Choose the numbers –1, –2, and –1/2, in that order.)

k < 0
Column A Column B

If k is –1 or –2, then Column A is larger since it is a product of squares. But if k = –1/2, then the two columns are equal:

Hence, there is not enough information to decide and the answer is (D).


Strategy: Sometimes you have to plug in the first three numbers (but never more than three) from a class of numbers.


x is both an integer and greater than 1. Let [x] stand for the smallest positive integer factor of x not equal to 1.
Column A Column B
[x] [x3]

Choose x = 2, 3, and 4. If x = 2, then [x] = 2 and [x3] = [8] = 2. So for this choice of x, the two columns are equal. If x = 3, then [x] = 3 and [x3] = [27] = 3, again the columns are equal.


Finally, If x = 4, then [x] = 2 and [x3] = [64] = 2, still again the columns are equal. Hence, the answer is (C).


Note, there is no need to check x = 5. The writers of the test do not change the results after the third number.

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