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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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