Inequality
Key Concepts & Formulas
| # | Concept | Quick Explanation |
|---|---|---|
| 1 | >, <, ≥, ≤, = | > : greater, < : smaller, ≥ : not smaller, ≤ : not bigger, = : equal |
| 2 | Chain Rule | Combine like signs in one direction: A>B>C ⇒ A>C; A<B<C ⇒ A<C |
| 3 | ≥ & ≤ in chain | Drop the equality for the ends: A≥B≥C ⇒ A≥C (A=C possible) |
| 4 | Either-or in conclusions | If two conclusions are A=B and A>B, and one must be true ⇒ “Either I or II” |
| 5 | Complimentary pairs | A≥B and A<B form a pair—exactly one is always true |
| 6 | Coded inequalities | Decode symbol first (e.g. © means >) then apply normal rules |
| 7 | Priority order | First decode, then make a single chain, then check every conclusion |
10 Practice MCQs
1. Statements: K ≥ L = M; N < L ≥ O. Conclusions: I. K > O II. M ≤ N
**Answer:** Only I is true **Solution:** Chain: K ≥ L (=M) > O ⇒ K > O (I true). M = L > N ⇒ M > N ⇒ II false. **Shortcut:** Draw single horizontal line: K---L(M)---O; signs face right. **Concept tag:** Chain rule, ≥ + > gives >2. Statements: A < B ≤ C = D; E ≥ C. Conclusions: I. E ≥ B II. D > A
**Answer:** Both I and II are true **Solution:** From B ≤ C ≤ E ⇒ E ≥ B (I). A < B ≤ C = D ⇒ A < D (II). **Shortcut:** Visualise two converging chains. **Concept tag:** Transitivity3. Statements: P > Q ≥ R = S; T ≥ Q. Conclusions: I. T ≥ R II. P > S
**Answer:** Both true **Solution:** T ≥ Q ≥ R ⇒ T ≥ R; P > Q ≥ R = S ⇒ P > S. **Concept tag:** Common middle term4. Statements: X ≤ Y ≤ Z; W ≥ Y > V. Conclusions: I. X ≤ W II. Z > V
**Answer:** Only I true **Solution:** X ≤ Y ≤ W ⇒ X ≤ W; Z ≥ Y > V ⇒ Z > V (but Y≤Z allows Z=V) so II not always true. **Shortcut:** Look for equality escape in ≥/≤. **Concept tag:** Equality trap5. Statements: 6 > 9 © 5 ▲ 3 (© = ≥, ▲ = <). Which expression is true?
**Answer:** 9 ≥ 5 **Solution:** Decode: 6>9, 9≥5, 5<3. Only 9≥5 is given in options. **Shortcut:** Replace symbols immediately. **Concept tag:** Coded inequality6. Statements: J ≤ K < L ≥ M; K ≥ N. Conclusions: I. J ≥ N II. L > N
**Answer:** Only II true **Solution:** J ≤ K ≥ N ⇒ no fixed relation (I false). K < L and K ≥ N ⇒ L > N (II true). **Shortcut:** Different-direction arrows ⇒ no conclusion. **Concept tag:** Opposite-direction rule7. Statements: A ≥ B ≥ C; D < C; E = B. Conclusions: I. A ≥ E II. D < A
**Answer:** Both true **Solution:** A ≥ B = E ⇒ A ≥ E; D < C ≤ B ≤ A ⇒ D < A. **Concept tag:** Equality substitution8. Statements: P ≤ Q < R; S ≥ R; T ≤ P. Conclusions: I. T < R II. S ≥ P
**Answer:** Both true **Solution:** T ≤ P ≤ Q < R ⇒ T < R; S ≥ R > Q ≥ P ⇒ S > P but still S ≥ P. **Shortcut:** ≥ includes equality—always safe. **Concept tag:** Safe conclusion9. Statements: M > N ≤ O = P; Q ≤ N. Conclusions: I. Q < M II. P ≥ Q
**Answer:** Only II true **Solution:** Q ≤ N < M ⇒ Q < M (I not must, Q=N possible). N ≤ O = P ⇒ Q ≤ P ⇒ P ≥ Q. **Concept tag:** Possible vs must be10. How many conclusions are definitely true? Statements: 8 ≤ 9 = 10 > 7; 6 < 8.
**Answer:** 2 **Solution:** 6<8≤9=10>7. Definite: 6<9, 8≤10. **Shortcut:** Count “must” arrows. **Concept tag:** Definite counting5 Previous Year Questions
1. Statements: A > B ≥ C; D ≤ C; E = B. Conclusions: I. E ≥ C II. A > D [RRB NTPC 2021]
**Answer:** Both I and II true **Solution:** E=B≥C ⇒ E≥C; A>B≥C≥D ⇒ A>D.2. Statements: P ≥ Q ≥ R; S < R; T ≥ P. Which is NOT true? [RRB Group-D 2019]
**Answer:** S ≥ T **Solution:** S3. If ▲ means >, ▼ means <, ● means =, which expression is correct? 7 ▲ 5 ● 5 ▼ 9 [RRB NTPC 2020]
**Answer:** 7 > 5 = 5 < 9 → 7 > 5 and 5 < 9 both valid.4. Statements: X ≥ Y = Z; W < Y; V ≥ X. Conclusions: I. V ≥ Z II. W < V [RRB ALP 2018]
**Answer:** Both true **Solution:** V≥X≥Y=Z ⇒ V≥Z; W5. Combined chain: 3 ≤ 4 < 5 ≥ 6 = 2. Which number is greatest? [RRB NTPC 2022]
**Answer:** 5 **Solution:** 5 lies at peak of ≥ and > signs.Speed Tricks & Shortcuts
| Situation | Shortcut | Example |
|---|---|---|
| 1 Single-direction chain | Cancel middle terms instantly | A>B>C>D ⇒ A>D |
| 2 ≥ chained with > | End letter gets strict inequality | A≥B>C ⇒ A>C |
| 3 Two conclusions A≥B & A<B | Mark “Either I or II” | No need to check further |
| 4 Coded symbols | Rewrite entire line with real symbols in 5 s | 5©7→5>7 |
| 5 Three-term loop | If directions clash ⇒ no relation | A>B<C gives no A-C link |
Common Mistakes to Avoid
| Mistake | Why Students Make It | Correct Approach |
|---|---|---|
| Ignoring equality escape | Assume ≥ always strictly greater | Remember A≥B allows A=B |
| Opposite-direction merge | Try to relate A-D in AC | Insert middle common term or say “can’t be determined” |
| Forgetting Either-or | Treat contradictory conclusions as both false | Recognise complementary pair and choose either |
| Mis-decoding coded symbols | Rush and swap ≥ with > | Double-code table in rough sheet first |
Quick Revision Flashcards
| Front | Back |
|---|---|
| Chain direction same | Combine fearlessly |
| ≥ + > | Results in > |
| A≥B and A<B | Exactly one true → Either-or |
| Coded © means | © = ≥ (typical code) |
| Opposite arrows | No conclusion possible |
| Equality in middle | Can be dropped at ends |
| Single peak rule | Greatest element sits at > / ≥ peak |
| “Only I true” shortcut | Check second with equality trap |
| Complementary pair examples | A≥B & A<B, A=B & A>B |
| Time per inequality Q | Aim ≤ 35 s |
BETA
