Inequalities
Here’s a list of most commonly used inequality signs:
| Inequality Sign | Symbol |
|---|---|
| Equal to | = |
| Less than | < |
| Greater than | > |
| Less than or Equal to | ≤ |
| Greater than or Equal to | ≥ |
Also, have a look at the following:
- A ≮ B means A ≥ B
- A ≰ B means A > B
- A ≯ B means A ≤ B
- A ≱ B means A < B
- A ≠ B means A < B or A > B
Let’s us see all the different types of questions that we may encounter in this chapter.
Types of Inequalities based Questions
Type 1: Finding relation between two variables in Simple Inequalities
Here, we will encounter two types of questions:
- Those having one statement
- Those having more than one statement – In such case we have to combine the inequalities wherever possible.
Two points to remember, or you can say the two steps involved in solving such questions:
- Step 1: Relation between two variables can only be determined if all the inequality signs are facing in the same direction.
- Step 2: < and > signs will take precedent over ≤ and ≥ signs, and these over the = sign.
Let’s see some examples …
Q. If A > B > C, then find the relation between A and C.
Explanation:
Step 1: Can we find relation between A and C?
All the inequality signs between A and C are facing in the same direction. It means we can find the relation between A and C.
Step 2: What is the relation between A and C?
There are two > signs between A and C. So, A > C.
Q. If A > B ≥ C, then find the relation between A and C.
Explanation:
Step 1: Can we find relation between A and C?
All the inequality signs between A and C are facing in the same direction. It means we can find the relation between A and C.
Step 2: What is the relation between A and C?
There is one > sign, and one ≥ sign between A and C. We know that, > will take precedence over ≥.
So, A > C.
Q. If A ≥ B < C, then find the relation between A and C.
Explanation:
Step 1: Can we find relation between A and C?
All the inequality signs between A and C are not facing in the same direction. It means we cannot find the relation between A and C.
It means:
- Case a: A > B < C OR
- Case b: A = B < C
Case b implies A < C, but case a implies A > C, A = C or A < C, i.e. cannot be determined.
Focus on the greater than and smaller than symbols between the variables being compared. A single relation can be determined only if all of their mouths face in the same direction.
If they face in different directions than either there will be more than one answer or the answer cannot be determined.
Q. Statements: H < J, F < H, I ≤ K = J
Conclusions:
I. H > I
II. I ≤ F
Give answer
(a) if only Conclusion I follows
(b) if only Conclusion II follows
(c) if either Conclusion I or II follows
(d) if neither Conclusion I nor II follows
(e) if both Conclusions I and II follow
Explanation:
Step 1: Combine all the statements
H < J, F < H, I ≤ K = J will become:
F < H < J = K and I ≤ K
Step 2: Check the validity of the given conclusions with the help of the newly derived statement(s):
I. H > I : Indefinite
II. I ≤ F : Indefinite
Answer: (d)
Type 2: Finding relation between two variables in Coded Inequalities
Unlike the simple inequalities, coded inequalities have all the signs (>, <, =, ≥, ≤) in coded form. i.e., substituted symbols are used in place of real symbols.
The candidates are required to replace the codes with real signs and then solve the questions in the same way as the questions of simple inequalities are solved.
Q. In the questions that follow, the symbols are used as follows.
A©B means A is greater than B.
A@B means A is either greater than or equal to B.
A#B means A is equal to B.
A&B means A is smaller than B.
A%B means A is either smaller than or equal to B.
Statements: P©Q, Q#R, S%R
Conclusions:
I. P@R
II. S&Q
Give answer
(a) if only Conclusion I follows
(b) if only Conclusion II follows
(c) if either Conclusion I or II follows
(d) if neither Conclusion I nor II follows
(e) if both Conclusions I and II follow
Explanation:
A©B means A is greater than B. (A©B → A > B)
A@B means A is either greater than or equal to B. (A@B → A ≥ B)
A#B means A is equal to B. (A#B → A = B)
A&B means A is smaller than B. (A&B → A < B)
A%B means A is either smaller than or equal to B. (A%B → A ≤ B)
Statements: P©Q: P > Q; Q#R: Q = R; S%R: S ≤ R
Conclusions:
I. P@R: P ≥ R
II. S&Q: S < Q
Step 1: Combine all the statements
On combining the three statements, we get: P > Q = R ≥ S
Step 2: Check the validity of the given conclusions with the help of the newly derived statement(s):
I. P@R: P ≥ R
It does not follow (P > Q = R, so P > R)
II. S&Q: S < Q
It does not follow (Q = R ≥ S, so Q ≥ S)
Answer: (d)
Some students find it difficult to understand the following option: “if either Conclusion I or II follows”.
It means that Exactly one Conclusion follows (either one may follow). If I follows than II doesn’t and vice-versa.
Type 3: Symbol Substitution
Q. If > denotes +, < denotes -, + denotes ÷, ^ denotes ×, - denotes =, × denotes >, and = denotes <, choose the correct statement from among the following:
(a) 13 > 7 < 6 + 2 = 3 ^ 4 (b) 9 > 5 > 4 - 18 + 9 > 16 (c) 9 < 3 < 2 > 1 × 8 ^ 2 (d) 28 + 4 ^ 2 = 6 ^ 4 + 2
(SSC CGL Question)
Explanation:
| Symbol | What it denotes |
|---|---|
| > | + |
| < | - |
| + | ÷ |
| ^ | × |
| - | = |
| × | > |
| = | < |
Option (a): 13 > 7 < 6 + 2 = 3 ^ 4
On substituting the symbols, we get:
13 + 7 - 6 ÷ 2 < 3 × 4
or 13 + 7 - 3 < 12
or 17 < 12 (not true)
Option (b): 9 > 5 > 4 - 18 + 9 > 16
On substituting the symbols, we get:
9 + 5 + 4 = 18 ÷ 9 + 16
or 18 = 2 + 16
or 18 = 18 (true)
Option (c): 9 < 3 < 2 > 1 × 8 ^ 2
On substituting the symbols, we get:
9 - 3 - 2 + 1 > 8 × 2
or 5 > 16 (not true)
Option (d): 28 + 4 ^ 2 = 6 ^ 4 + 2
On substituting the symbols, we get:
28 ÷ 4 × 2 < 6 × 4 ÷ 2
or 7 × 2 < 6 × 2
or 14 < 12 (not true)
Answer: (b)