⭐ Sabse Important Points · एक नज़र में
- Symbols: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to), = (equal to), ≠ (not equal to).
- Rule of thumb: For a definite conclusion between two elements, the inequality signs must point in the same direction when combined (e.g., A > B > C gives A > C). If signs are opposite (A > B < C), no relation between A and C can be established.
- Combining inequalities: If A > B and B > C, then A > C (transitive property). Similarly, if A ≥ B and B ≥ C, then A ≥ C.
- Either-or case: If two conclusions are individually false but together cover all possibilities (e.g., A ≥ B and A < B), then either one follows. This occurs when elements are complementary.
- Priority of symbols: '>' and '<' are strict, while '≥' and '≤' include equality. When combining, the strict inequality dominates if both are present (e.g., A > B ≥ C gives A > C).
- Checking conclusions: A conclusion follows only if it is true in all possible scenarios. If there is any ambiguity, it does not follow.
Introduction · परिचय
Inequalities in Reasoning Ability test the ability to compare quantities using symbols like >, <, ≥, ≤, =, and ≠. Questions typically present statements and conclusions, requiring you to determine which conclusions logically follow. This topic is scoring if you master the basic rules and avoid common traps.
Basic Inequality Symbols and Their Meanings
Inequalities use symbols to show relationships between quantities. The symbol '>' means 'greater than', '<' means 'less than', '≥' means 'greater than or equal to', '≤' means 'less than or equal to', '=' means 'equal to', and '≠' means 'not equal to'. In reasoning questions, you often need to combine statements to derive conclusions.
⭐ Key Points · याद रखने वाली बातें
- Symbols: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to), = (equal to), ≠ (not equal to).
- Combining inequalities: If A > B and B > C, then A > C (transitive property). Similarly, if A ≥ B and B ≥ C, then A ≥ C.
- Priority of symbols: '>' and '<' are strict, while '≥' and '≤' include equality. When combining, the strict inequality dominates if both are present (e.g., A > B ≥ C gives A > C).
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Statements: P > Q, Q ≥ R. Conclusions: I. P > R, II. P ≥ R.
✅ Solution: Only I follows.
💡 Explanation: From P > Q and Q ≥ R, we get P > R (since Q ≥ R means Q is at least R, and P is greater than Q, so P is strictly greater than R). Conclusion II (P ≥ R) is also true but not necessarily; however, in such questions, if P > R is true, then P ≥ R is also true, but typically only the strongest conclusion is considered. Actually, both I and II follow because P > R implies P ≥ R. But in many exams, they ask which is true; here both are true, but usually only one is asked. For this example, both follow.
Combining Inequalities and Deriving Conclusions
To solve inequality questions, combine the given statements by aligning common terms. The key is to ensure the common term is in the middle and the direction of inequality is consistent. Then derive a relationship between the outer terms. If the direction conflicts, no definite conclusion can be drawn.
📐 Formula / Rule Box · सूत्र याद रखो
Rule of thumb
For a definite conclusion between two elements, the inequality signs must point in the same direction when combined (e.g., A > B > C gives A > C). If signs are opposite (A > B < C), no relation between A and C can be est
⭐ Key Points · याद रखने वाली बातें
- Either-or case: If two conclusions are individually false but together cover all possibilities (e.g., A ≥ B and A < B), then either one follows. This occurs when elements are complementary.
- Checking conclusions: A conclusion follows only if it is true in all possible scenarios. If there is any ambiguity, it does not follow.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Statements: A ≥ B > C, C = D ≤ E. Conclusions: I. A > D, II. A ≥ E.
✅ Solution: Only I follows.
💡 Explanation: From A ≥ B > C and C = D, we get A > D (since A ≥ B > C = D implies A > D). For II, A ≥ B > C = D ≤ E gives no definite relation between A and E because the chain breaks at D ≤ E (opposite direction). So II does not follow.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Statements: M > N, N ≤ O, O < P. Conclusions: I. M > O, II. P > N.
👀 Answer dekho
✅ Answer: Only II follows.
💡 Explanation: From M > N and N ≤ O, we cannot conclude M > O because N ≤ O means N could be equal to O, and M > N does not guarantee M > O (if N=O, M>N implies M>O, but if N<O, M could be less than O). So I does not follow. From N ≤ O and O < P, we get N < P (since N ≤ O < P implies N < P). So II follows.
Q2Statements: X = Y, Y ≥ Z, Z > W. Conclusions: I. X > W, II. X ≥ Z.
👀 Answer dekho
✅ Answer: Both I and II follow.
💡 Explanation: From X = Y and Y ≥ Z, we get X ≥ Z (II follows). From X ≥ Z and Z > W, we get X > W (I follows). So both follow.
Q3Statements: A < B, B ≤ C, C = D. Conclusions: I. A < D, II. A ≥ D.
👀 Answer dekho
✅ Answer: Only I follows.
💡 Explanation: From A < B ≤ C = D, we get A < D. So I follows. II says A ≥ D, which is false. Hence only I follows.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Assuming a definite relation when signs are opposite: For example, from A > B < C, do not conclude A > C or A < C; no relation can be established.
- Ignoring the 'either-or' case: When two conclusions are complementary (e.g., A ≥ B and A < B), one of them must be true, but individually they may not follow. Always check if they cover all possibilities.
- Misinterpreting '≥' and '≤': Remember that '≥' means greater than or equal to, so it includes the possibility of equality. When combining with '>', the strict inequality dominates (e.g., A > B ≥ C gives A > C).
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Transitive property: If A > B and B > C, then A > C. This holds for all inequality symbols except when directions conflict.
- Either-or case: If two conclusions are complementary (e.g., A ≥ B and A < B), then either one follows, provided no other relation is possible.
- Checking conclusions: A conclusion is true only if it holds in every possible scenario consistent with the statements. If there is any doubt, it does not follow.