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Inequalities

Inequalities

MODULES

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Basics of Inequalities
Quadratic Inequalities
Basics of Modulus
Multiple Modulus Functions
Sum or Product is Constant
Max & Min for Range & Substitution
Past Questions

CONCEPTS & CHEATSHEET

Concept Revision Video

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Inequalities 1
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Inequalities 2
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PRACTICE

Inequalities : Level 1
Inequalities : Level 2
Inequalities : Level 3
ALL MODULES

CAT 2025 Lesson : Inequalities - Basics of Inequalities

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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>ba \gt ba>b aaa is greater than bbb
a<ba \lt ba<b aaa is less than bbb
a≥ba \ge ba≥b aaa is greater than or equal to bbb
a≤ba \le ba≤b aaa is less than or equal to bbb
a≠ba \ne ba=b aaa is not equal to bbb


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
kkk is any real number, adding/subtracting kkk 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
kkk is a real number and k>0\bm{k \gt 0}k>0, then multiplying/dividing kkk on both sides of an inequation leaves it unchanged. However, if k<0\bm{k \lt 0}k<0, then multiplying/dividing kkk 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>ba \gt ba>b,
k∈rk \in rk∈r
a+k>b+ka + k \gt b + ka+k>b+k
a−k>b−ka - k \gt b - k a−k>b−k
a−b>0a - b \gt 0a−b>0 or 0>b−a0 \gt b - a0>b−a
If x>5+yx \gt 5 + yx>5+y ⇒ x−y>5x - y \gt 5x−y>5
⇒
−5>y−x -5 \gt y - x−5>y−x ⇒ y−x<−5y - x \lt -5y−x<−5
a>ba \gt ba>b
k>0,j<0k \gt 0, j \lt 0k>0,j<0
ak>bkak \gt bkak>bk
aj<bjaj \lt bjaj<bj
2x−10>22x - 10 \gt 22x−10>2 ⇒ x−5>1x - 5 \gt 1x−5>1
a−b>−2a - b \gt -2a−b>−2 ⇒ a−b−2<1\dfrac{a - b}{-2} \lt 1−2a−b​<1 ⇒ b−a2<1\dfrac{b - a}{2} \lt 12b−a​<1


Note:
a>ba \gt ba>b cannot be written as ab>1\dfrac{a}{b} \gt 1ba​>1 as we do not know if the variable bbb 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

If 5−2x>75-2x \gt 75−2x>7, then which of the following is true?
(111) x>1x \gt 1x>1            (222) x<1x \lt 1x<1            (333) x>−1x \gt -1x>−1            (444) x<−1x \lt -1x<−1           

Solution

To find the range of
xxx, keep the variable on one side and move the constants to the other.

5−2x>7 5 - 2x \gt 7 5−2x>7

⇒
5−7>2x5 - 7 \gt 2x 5−7>2x

⇒
−22>x \dfrac{-2}{2} \gt x 2−2​>x

⇒
−1>x -1 \gt x −1>x

⇒
x<−1 x \lt -1 x<−1

Answer: (
444) x<−1 x \lt -1 x<−1


Example 2

If 4x−13≤3 4x-13 \le 3 4x−13≤3 and 2x−1≥1 2x-1 \ge 1 2x−1≥1, then which of the following is an acceptable range of xxx?
(111) [1,3][1,3][1,3]            (222) [1,4][1,4][1,4]            (333) [−5,3][-5,3][−5,3]            (444) [3,4][3,4][3,4]           

Solution

4x−13≤34x-13 \le 34x−13≤3

⇒
4x≤164x \le 164x≤16

⇒
x≤4x \le 4x≤4 -----(1)

2x−1≥12x-1 \ge 12x−1≥1

⇒
2x≥22x \ge 22x≥2

⇒
x≥1x \ge 1x≥1-----(2)

Combining the conditions (1) and (2), we get
1≤x≤41 \le x \le 41≤x≤4.

Answer: (
222) [1,4][1,4][1,4]


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