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 – $$|x|=\begin{cases}x, \hspace{0.9cm} if \ x \ge 0 \ and \\ -x, \hspace{0.5cm} if \ x < 0 \end{cases} $$
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.