I won't give a specific question as an example; I'm sure everyone knows what I mean by "inequality/greater than/less than" questions. Speaking generally, they're the "is X greater than such-and-such..."
The answer, of course, can be tricky, because X could be positive, negative, zero, or a fraction, decimal, complex fraction, etc.
So, what are the best "plug-ins" to use to hypothesize what X could be? The superficial test-prep books, of course, advise students to try three plug-ins: a positive number, a negative number, and a zero. But that's often an inadequate range of plug-ins for complex quant questions.
Can anyone suggest a more comprehensive (yet still simple and limited) range of plug-ins to use in these situations?
Much thanks!
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Best "plug-ins" for inequality questions?
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Geva@EconomistGMAT
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I usually go with 2, 1/2, -1/2, -2 as representative of their respective "regions" of the number line. Unless the problem, indicates somehow that some other number is better. It really does come down to what the problem needs - what issue is tested (pos. Vs. negative Vs. Zero, integer Vs fraction, and numerous other cases).yates wrote:I won't give a specific question as an example; I'm sure everyone knows what I mean by "inequality/greater than/less than" questions. Speaking generally, they're the "is X greater than such-and-such..."
The answer, of course, can be tricky, because X could be positive, negative, zero, or a fraction, decimal, complex fraction, etc.
So, what are the best "plug-ins" to use to hypothesize what X could be? The superficial test-prep books, of course, advise students to try three plug-ins: a positive number, a negative number, and a zero. But that's often an inadequate range of plug-ins for complex quant questions.
Can anyone suggest a more comprehensive (yet still simple and limited) range of plug-ins to use in these situations?
Much thanks!
My best general advice is to start with the easiest plug in you can think of (that satisfies the question stem) and see what answer you get, then stop and think what plug in might actually bring you the opposite result. At the end of the day, plugging in is a tool for elimination, not for proving something. For example, if the problem asks "is X>5?", start with the easiest plug in that satisfies the statmeents (let's say that the statements allow x=10). Then spend the time thinking whether you can find the opposite result, i.e. a value of x that is less than 5. This could be a fraction, a negative, zero, one, doesn't matter - you just need something less than 5 to get the opposite answer of "no".
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Ian Stewart
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It really depends on the type of question you're looking at. A lot of different situations can crop up in GMAT inequality questions, so it's impossible to list any number-picking rules that will be applicable to every question. In general, most inequality questions hinge on negatives and positives. So when plugging in numbers, you normally want to pick numbers that will make any terms in a product or fraction positive and negative. For example, if you see this inequality:
(x-11) / (x-12) < 0
plugging in one positive value and one negative value for x is not going to tell you anything useful. Instead you want to plug in values that might make the numerator positive or negative, and the denominator positive or negative. Here the relevant numbers are 11 and 12, so you should plug in one number less than 11, one number between 11 and 12, and one number greater than 12. Then you'll see that the inequality above can only be true when 11 < x < 12.
Further, when you have inequalities with simple powers of x on both sides, we normally need to consider not only positives and negatives, but also fractions near zero (both positive and negative). That is because values near zero behave differently from whole numbers when we raise them to exponents. For example, if x = 2, and we then cube x (that is, find x^3), we get a larger value. But if x = 1/2 and we cube x, we get a smaller value. The same occurs with negatives; if x = -2, and we cube x, then we get a smaller value, but if x = -1/2 and we cube x, we get a larger value, since -1/8 is larger than -1/2. So if you see an inequality with simple powers of x on both sides, like the following:
x^3 > x
then you'd normally want to test four values: one value less than -1, one value between -1 and 0, one value between 0 and 1, and one value greater than 1. You then find that the above inequality is true when -1 < x < 0, and when x > 1.
I think a lot of test takers end up in 'traps' when they use number-picking for inequality questions. It can be a reliable strategy provided you know exactly which numbers to test. But if you miss one exceptional case, you can often get the wrong answer by number-picking. If you understand the algebraic rules for manipulating inequalities, those can often more reliably (and more quickly) get you to the right answer.
(x-11) / (x-12) < 0
plugging in one positive value and one negative value for x is not going to tell you anything useful. Instead you want to plug in values that might make the numerator positive or negative, and the denominator positive or negative. Here the relevant numbers are 11 and 12, so you should plug in one number less than 11, one number between 11 and 12, and one number greater than 12. Then you'll see that the inequality above can only be true when 11 < x < 12.
Further, when you have inequalities with simple powers of x on both sides, we normally need to consider not only positives and negatives, but also fractions near zero (both positive and negative). That is because values near zero behave differently from whole numbers when we raise them to exponents. For example, if x = 2, and we then cube x (that is, find x^3), we get a larger value. But if x = 1/2 and we cube x, we get a smaller value. The same occurs with negatives; if x = -2, and we cube x, then we get a smaller value, but if x = -1/2 and we cube x, we get a larger value, since -1/8 is larger than -1/2. So if you see an inequality with simple powers of x on both sides, like the following:
x^3 > x
then you'd normally want to test four values: one value less than -1, one value between -1 and 0, one value between 0 and 1, and one value greater than 1. You then find that the above inequality is true when -1 < x < 0, and when x > 1.
I think a lot of test takers end up in 'traps' when they use number-picking for inequality questions. It can be a reliable strategy provided you know exactly which numbers to test. But if you miss one exceptional case, you can often get the wrong answer by number-picking. If you understand the algebraic rules for manipulating inequalities, those can often more reliably (and more quickly) get you to the right answer.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com
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