Numbers: Characteristics
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1. If x3 < x < x2 then:

(A) 0 < x < 1
(B) 1 < x < 2
(C) –1 < x < 0
(D) x < –1
(E) x > 2

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Positives bigger than 1 get bigger when you square or cube them. Negative numbers with absolute values bigger than 1 get smaller when you cube them and bigger when you square them. Positives between 0 and 1 get smaller when you square them and even smaller when you cube them. Negatives between 0 and –1 get a lot bigger when you square them (they become positive) and a little bigger when you cube them (they stay negative but get closer to zero. In this case x is a number like –2 (gets bigger if you square it and smaller if you cube it). So the answer is (D).

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2. If x < x3 < x2 then:

(A) 0 < x < 1
(B) 1 < x < 2
(C) –1 < x < 0
(D) x < –1
(E) x > 2

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Using the discussion for problem 1, x must be a negative fraction like –1/2 that gets a lot bigger when you square it (becomes +1/4) and a little bigger when you cube it (becomes –1/8). So the answer is (C).

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3. If x > y and x ≠ 0 and y ≠ 0 then:

(A) x2 > y2
(B) 1/x > 1/y
(C) 1/x < 1/y
(D) x3 > y3
(E) none of these

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Choice A is wrong if x = 1 and y = –100. Choice B is wrong if x = 100 and y = 1. Choice C is wrong if x = 100 and y = –1. Cubing fractions (with absolute values less than 1) moves them closer to zero. Cubing numbers with absolute values greater than 1 moves them away from zero. On a number line, if x is to the right of y (that is if x is greater than y) then x3 will still be to the right of y3 no matter what the original numbers were. So the answer is (D).

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4. If x + 2y > 2x – 3y then which of the following is true?

I. The value of x/y must be less than 5.
II. The value of x could be greater than the value of y.
III. If y < 0 then x < 0.

(A) II, only
(B) III, only
(C) I and II, only
(D) II and III, only
(E) I and III, only

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If you rearrange the equation you get 5y>x. Now you divide both sides by y, but you have to be careful. If y is positive then you get x/y<5. If y is negative then you get x/y>5 because when you divide by a negative you switch the direction of the inequality. So (I) is not true. However, (II) could be true because x=4, y=1 would be okay according to x/y<5 (y>0). And (III) is true because x/y>5 (y<0) forces x to be negative when y is negative (otherwise x/y couldn't be bigger than +5). So (II) and (III) are true and the answer is (D).

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5. If r < s < t then which of the following must be true?

I. rs < st
II. |1/r| > |1/s|
III. s2 > r

(A) none
(B) I, only
(C) II, only
(D) III, only
(E) I and III, only

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If r and s are both negative then (I) is not true. If r = –100 and s = 1, then (II) is not true and neither is (III). So the answer is (A).

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6. If y = |1/(2–x)| and x≠2 then y increases when:

(A) x increases from 100 to 102
(B) x decreases from –100 to –102
(C) x decreases from 2.3 to 2.1
(D) x decreases from 1.7 to 1.5
(E) x decreases from 0 to –2

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To make y big you want the denominator to be small. That means you want x to be close to 2 (above or below doesn't matter). In choice (C), x gets closer to 2 which makes y bigger. So the answer is (C).

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7. If a is an integer and b=a3 then which of the following could be equal to √b?

(A) 27
(B) 21
(C) 15
(D) 9
(E) 3

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This problem looks much harder than it really is. Let's be concrete. The value of b is either 272, 212, 152, 92, or 32. But the cube root of b has to be the integer, a. Only one of the numbers in this list will come out to an integer if you cube root it and that's 272 because the cube root of 27 is 3 so the cube root of 272 is 32 or 9. The answer is (A).

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8. Which of the following is true?

I. 550 is even
II. 550 + 749 is odd
III. 3 · (833) is odd

(A) none
(B) I, only
(C) I and II, only
(D) II and III, only
(E) I and III, only

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The product of a long string of odd numbers is still odd because there are no 2's. So (I) is not true. The sum of two odd numbers is even so (II) is not true. If the product of a long string of numbers has one or more 2's then it is even. So (III) is not true either and the answer is (A).

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9. If j and k are integers and j + 4 = 2k then which of the following must be true?

I. j is even
II. k is odd
III. (j+k) is even

(A) none
(B) I, only
(C) II, only
(D) I and II, only
(E) I, II, and III

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We know that j+4 is even because it is equal to 2k and 2k is definitely even. This means j must be even (if j were odd, then j+4 would be odd because even plus odd is odd). We don't know anything about k, it could be odd or even so we don't know anything about j+k, it could also be odd or even. So only (I) is true and the answer is (B).

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10. If x is an odd integer and z = x2+4x+4 then which of the following is true?

I. z cannot be prime
II. z cannot be negative
III. √z is an integer

(A) none
(B) I, only
(C) II, only
(D) III, only
(E) I, II, and III

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The quadratic factors to (x+2)·(x+2). This means z is a perfect square which means it isn't prime (it has at least two factors) and isn't negative. Of course, the square root of z is (x+2) and is an integer since x is an integer. So (I), (II), and (III) are all true and the answer is (E).

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