This is part of our
GRE Math Essentials project is the best complement to our
Official GRE Quant Book. It provides a cutting-edge, in-depth overview of all the math concepts to achieve 170 in the Quantitative Reasoning portion of the GRE. The book still remains our hallmark. However, the following chapters will give you a lot of tips, tricks, and shortcuts to make your quant preparation more robust and solid.
Let a and b be real numbers. If a – b is negative we say that a is less than b and write a < b. If a – b is positive then a is greater than b, i.e., a > b.
1. For any two real numbers a and b, we have a > b or a = b or a < b.
2. If a > b and b > c, then a > c. If a > b then (a + c) > (b + c) and (a - c) > (b - c), however, ac > bc and (a/c) > (b/c) (not sure) (is true only when c is positive)
3. If a > b, then a + m > b + m, for any real number m.
4. If a ≠ 0, b ≠ 0 and a > b, then \(\frac{1}{a} < \frac{1}{b}\).
5. If a > b, then am > bm for m > 0 and am < bm for m < 0,
that is, when we multiply both sides of inequality by a negative quantity, the sign of inequality is reversed.
6. If a > X, b > Y, c > Z then (1) a + b + c + .... > X + Y + Z + .... (2) abc .... > XYZ .... (Provided none is negative)
7. If x > 0 and a > b > 0, then \(a^x > b^x\)
8. If a > 1 and x > y > 0, then \(a^x > a^y\)
9. If 0 < a < 1 and x > y > 0, then \(a^x < a^y\)
10. Do not cancel anything from both sides of inequality unless you are sure that the canceled quantity is positive, so ax > ay does not necessarily mean x > y, etc.
11. The concept of number line is very useful in checking inequalities. The common values to check are x = 0, 1, -1, >1 (preferred value = 2), between 0 and 1 (preferred value = 1/2), between - 1 and 0 (preferred value = -1/2), and less than -1 (preferred value = -2). So in short, there are 7 points: -2, -1, -1/2, 0, 1/2, 1, 2.
12.
|x| is defined as the non-negative value of x and hence is never negative. On the GRE, \(\sqrt{x^2} = |x|\), that means, the square root of any quantity is defined to be non-negative, so \(\sqrt{36} = 6\) and not − 6 on the GRE. BUT if \(x^2=36\) ⇒ \(x=6\) or \(-6\) both. So \(\sqrt{x^2}= x\) or \(-x\) both are possible. If, x is negative, then \(\sqrt{x^2}=-x\) as it has to be +ve eventually. In this case x is negative and -x is positive.13. |5| = 5, |-5| = 5, so |x| = x, if x is positive or 0 and |x| = -x if x is negative.
14. If |x| > x, then x is negative.
15. If |x| = a, then x = a or x = -a.
16. If |x| > a, then x > a or x < -a.
17. If |x| < a, then x < a or x > -a.
18. If |x - a| > b, then either x - a > b or x - a < -b
19. If |x - a| < b, then either x - a < b or x - a > -b.
20. If |x| = x, then x is either positive or 0.
21. |a + b| ≤ |a| + |b|, |a – b| ≥ ||a| – |b||, |ab| = |a| |b|, \(|\frac{a}{b}|=|\frac{a}{b}|\), \(b \neq 0\), \(|a^2|=a^2\)
22. If (x - a) (x - b) < 0, then x lies between a and b. OR a < x < b.
23. If (x - a) (x - b) > 0, then x lies outside a and b. OR x < a, x > b.
24. If \(x^2> x\), then either x > 1 or x is negative (x < 0)
25. If \(x^2< x\), then x lies between 0 and 1. (0 < x < 1)
26. If \(x^2= x\), then x = 0 or x = 1.
27. If \(x^3> x\), then either x > 1 or x is between -1 and 0(either x > 1 or -1 < x < 0).
28. If \(x^3< x\), then either x lies between 0 and 1 or x is less than -1. (either 0 < x < 1 or x < -1)
29. If \(x^3= x\), then x = 0 or x = 1 or x = -1.
30. If \(x^3= x\), then x = 0 or x = 1 or x = -1.
31. If x > y, it is not necessary that \(x^2> y^2\) or \(\sqrt{x} > \sqrt{y}\) etc. So even powers can’t be predicted.
32. If x > y, it is necessarily true that \(x^3 > y^3\) or \(\sqrt[3]{x} > \sqrt[3]{y}\) etc. So odd powers and roots dont change sign.
33. ab > 0 means \(\frac{a}{b} > 0\) and vice versa. The two are of the same sign.
34. ab < 0 means \(\frac{a}{b} < 0\) and vice versa. The two are of the opposite sign.
35. If x is positive, \(x + \frac{1}{x} ≥ 2\) .
36. If X is positive, then
(
1) \(\frac{(a + X) }{ (b + X)} > \frac{a}{b}\) if a < b
(
2) \(\frac{(a + X) }{ (b + X)} < \frac{a}{b}\) if a > b
37. If X is negative, then
(
1) \(\frac{(a + X) }{ (b + X)} > \frac{a}{b}\) if a > b
(
2) \(\frac{(a + X) }{ (b + X)} < \frac{a}{b}\) if a < b
38. \(\frac{(a + c + e + ....) }{ (b + d + f + ....)}\) is less than the greatest and greater than the least of the fractions \(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}, .....
\)
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