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Inequalities

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CAT 2025 Lesson : Inequalities - Basics of Inequalities

Inequalities simply represent the relationship between two quantities that are not equal to one another. Inequalities arise when one expression is greater than or less than another expression. ie. A

\gt

B or A

\ge

B and B

\lt

A or B

\le

A. In CAT and other entrance tests, about 2 to 3 questions in inequalities appear every year.

1. Symbols & Representations

The basic types of inequalities between two items are enumerated below.

Representation	Meaning
$a \gt b$	$a$ is greater than $b$
$a \lt b$	$a$ is less than $b$
$a \ge b$	$a$ is greater than or equal to $b$
$a \le b$	$a$ is less than or equal to $b$
$a \ne b$	$a$ is not equal to $b$

Common ways of writing the range of values where the inequalities hold good are as follows.

Representation	Meaning
(3, 4)	All real numbers between 3 and 4, but not including 3 and 4.
[3, 4]	All real numbers between 3 and 4, including 3 and 4.
[3, 4)	All real numbers between 3 and 4, including 3 but not including 4.
(3, 4]	All real numbers between 3 and 4, not including 3 but including 4.

2. Arithmetic Operations on Inequalities

Addition and Subtraction: Where

k

is any real number, adding/subtracting

k

on both sides of an inequation leaves it unchanged. We can move variables from one side of the inequation to the other using addition/subtraction.

Multiplication & Division: Where

k

is a real number and

\bm{k \gt 0}

, then multiplying/dividing

k

on both sides of an inequation leaves it unchanged. However, if

\bm{k \lt 0}

, then multiplying/dividing

k

on both sides of an inequation, changes the sign of the inequality. We cannot move variables from one side of the inequation to the other using multiplication/division.

If	then	Example
$a \gt b$ , $k \in r$	$a + k \gt b + k$ $a - k \gt b - k$ $a - b \gt 0$ or $0 \gt b - a$	If $x \gt 5 + y$ ⇒ $x - y \gt 5$ ⇒ $-5 \gt y - x$ ⇒ $y - x \lt -5$
$a \gt b$ $k \gt 0, j \lt 0$	$ak \gt bk$ $aj \lt bj$	$2x - 10 \gt 2$ ⇒ $x - 5 \gt 1$ $a - b \gt -2$ ⇒ $\dfrac{a - b}{-2} \lt 1$ ⇒ $\dfrac{b - a}{2} \lt 1$

Note:

a \gt b

cannot be written as

\dfrac{a}{b} \gt 1

as we do not know if the variable

b

is negative or positive.

3. Linear Inequalities

Most linear inequalities will involve 1 variable. We will be required to find the range or number of integral solutions that exist for a given inequality.

These involve 1 variable with the highest power being 1. Direct questions of this form are uncommon as these are very simple. However, to solve higher order inequalities (like quadratic, cubic, etc.) or inequalities with modulus we reduce them to simple linear inequalities.

Example 1

5-2x \gt 7

, then which of the following is true?
(

1

)

x \gt 1

(

2

)

x \lt 1

(

3

)

x \gt -1

(

4

)

x \lt -1

Solution

To find the range of

x

, keep the variable on one side and move the constants to the other.

5 - 2x \gt 7

⇒

5 - 7 \gt 2x

⇒

\dfrac{-2}{2} \gt x

⇒

-1 \gt x

⇒

x \lt -1

Answer: (

4

)

x \lt -1

Example 2

4x-13 \le 3

and

2x-1 \ge 1

, then which of the following is an acceptable range of

x

?
(

1

)

[1,3]

(

2

)

[1,4]

(

3

)

[-5,3]

(

4

)

[3,4]

Solution

4x-13 \le 3

⇒

4x \le 16

⇒

x \le 4

-----(1)

2x-1 \ge 1

⇒

2x \ge 2

⇒

x \ge 1

-----(2)

Combining the conditions (1) and (2), we get

1 \le x \le 4

.

Answer: (

2

)

[1,4]

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