CAT Quant : Inequalities and Modulus – Important Formulas and Concepts

Basics

  • If ‘a’ is any real number, then ‘a’ is either positive or negative or zero.
    • When ‘a’ is positive, we write a>0 which is read ‘a is greater than zero’.
    • When ‘a’ is negative, we write a<0 which is read ‘a is less than zero’.
    • If ‘a’ is zero, we write a=0 and in this case, ‘a’ is neither positive nor negative.
  • For any two non-zero real numbers a and b –
    • a is said to be greater than b when a-b is positive, written as a>b when a-b > 0
    • a is said to be less than b when a-b is negative, written as a<b when a-b < 0

Inequalities

  • Inequalities compare two quantities or expressions and describe their relationship.
  • Common inequality symbols include: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

Properties of Inequalities

  • For any two real numbers a and b,
    • either a>b or a<b or a=b
    • If a > b, then b<a
    • If a < b, then a \( \ge \) b and if a>b , then a \( \le \) b
    • If a > b and b > c, then a > c
    • If a < b and b < c, then a < c
    • a > b, then a \( \pm \) c > b \( \pm \) c
    • If a > b and c > 0, then ac > bc
    • If a < b and c > 0, then ac < bc
    • If a > b and c < 0, then ac < bc
    • If a < b and c < 0, then ac > bc
    • If a > b and c > d, then a+c > b+d
    • If a < b and c < d, then a+c < b+d
    • If a > 0, then -a < 0 and if a > b then -a < -b
  • The square of any real number is always greater than or equal to 0
  • The square of any non-zero real number is always greater than 0
  • If a and b are positive numbers and a > b, then
    • \( \frac{1}{a} < \frac{1}{b} \)
    • \( \frac{a}{c} > \frac{b}{c} \) if c > 0
    • \( \frac{a}{c} < \frac{b}{c} \) if c < 0
  • For any two positive numbers a and b,
    • If a > b, then a2 > b2
    • If a2 > b2, then a > b
    • If a > b, then for any positive value of n, an > bn
  • Let, A, G and H be the Arithmetic mean, Geometric mean and Harmonic mean of n positive real numbers. Then \( A \ge G \ge H \)< the equality occurring only when the numbers are all equal.
  • If the sum of two positive quantities is given, their product is the greatest when they are equal; and if the product of two positive quantities is given, their sum is the least when they are equal.
  • If a > b and c > d, then we cannot say anything conclusively about the relationship between (a-b) and (c-d); depending on the values of a, b, c and d, it is possible to have (a-b) > (c-d) or (a-b) = (c-d) or (a-b) < (c-d)
  • When two numbers a and b have to be compared, we can use one of the following two methods –
    • If both a and b are positive, we can take the ratio \{ \frac{a}{b} \} and depending on whether \{ \frac{a}{b} \} is less than, equal to or greater than 1, we can conclude that “a” is less than, equal to or greater than “b”.
    • In other words, for two positive numbers a and b,
      • if \{ \frac{a}{b} \} < 1, then a < b
      • if \{ \frac{a}{b} \} = 1, then a = b
      • if \{ \frac{a}{b} \} > 1, then a > b
    • If one or both of a and b are not positive or we do not know whether they are positive, negative or zero, then we can take the difference of a and b and depending on whether (a-b) is less than, equal to or greater than zero, we can conclude that “a” is less than, equal to or greater than “b”.
    • In other words, for any two real numbers a and b,
      • if (a-b) < 0, then a < b
      • if (a-b) = 0, then a = b
      • if (a-b) > 0, then a > b
  • For any positive number x \( \ge \) 1, \( \\2 \le {\LARGE(} 1+\frac{1}{x}{\LARGE)}^{x} < 2.8 \\ \)The equality in the first part will occur only if x = 1.
  • For any positive number, the sum of the number and its reciprocal is always greater than or equal to 2, i.e., \( \\ x+ \frac{1}{x} \ge 2 \hspace{1cm}where \ x > 0 \)The equality in this relationship will occur only when x = 1.

Absolute Value:

  • The modulus, or absolute value, of a real number x, represents the distance of x from zero on the number line.
  • written as |x| and read as “modulus of x”
  • For any real number x, the absolute value is defined as follows – $$|x|=\begin{cases}x, \hspace{0.9cm} if \ x \ge 0 \ and \\ -x, \hspace{0.5cm} if \ x < 0 \end{cases} $$

Properties of Modulus

  • For any real number x and y –
    • x=0 \( \Leftrightarrow \) |x|=0
    • |x|\ \( ge \) 0 and -|x| \( \le \) 0
    • |x+y| \( \le \) |x|+|y|
    • ||x|-|y|| \( \le \) |x-y|
    • -|x| \( \le \) x \( \le \) |x|
    • |x \( \cdot \) y| = |x| \( \cdot \) |y|
    • | \( \frac{x}{y} \)| = \( \frac{|x|}{|y|} \) ; (y \( \ne \) 0)
    • |x|2 = x2

Interval Notations

  • Interval notation is a concise way to represent sets of real numbers using brackets and parentheses. Some commonly used interval nottations are as follows
    • Closed Interval:
      • A closed interval includes its endpoints.
      • Denoted by square brackets.
      • Example: [a, b] includes all real numbers x such that a ≤ x ≤ b.
    • Open Interval:
      • An open interval excludes its endpoints.
      • Denoted by parentheses.
      • Example: (a, b) includes all real numbers x such that a < x < b.
  • Half-Open or Half-Closed Interval:
    • One endpoint is included, and the other is excluded.
    • Denoted by a combination of brackets and parentheses.
    • Example: [a, b) includes all real numbers x such that a ≤ x < b.
  • Unbounded Intervals:
    • An interval that extends indefinitely in one or both directions.
    • Denoted by the symbols ∞ (infinity) or −∞ (negative infinity).
    • Examples: (a, ∞) includes all real numbers greater than or equal to a, and (−∞, b] includes all real numbers less than or equal to b.
  • Singleton Interval:
    • An interval containing only one specific value.
    • Denoted by a single number within square brackets.
    • Example: [a, a] represents the set containing only the number a.

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