Equations & Inequations - Theory & Concepts
⚖️ Equations & Inequations - Complete Theory
Master algebraic equations - crucial for IBPS Mains!
🎯 What are Equations & Inequations?
Equation: Mathematical statement where two expressions are equal.
Example: 2x + 5 = 13
Inequation (Inequality): Mathematical statement comparing two expressions.
Example: 2x + 5 > 13 Symbols: > (greater than), < (less than), ≥ (greater than or equal), ≤ (less than or equal)
📐 Linear Equations
One Variable Linear Equation
Form: ax + b = c
Steps to solve:
- Move constants to one side
- Move variables to other side
- Isolate variable
Example:
3x + 7 = 22
3x = 22 - 7 3x = 15 x = 5
Two Variable Linear Equations
System of two equations:
a₁x + b₁y = c₁ … (1) a₂x + b₂y = c₂ … (2)
Methods to Solve:
Method 1: Substitution
Example: 2x + y = 10 … (1) x - y = 2 … (2)
From (2): x = y + 2
Substitute in (1): 2(y + 2) + y = 10 2y + 4 + y = 10 3y = 6 y = 2
x = 2 + 2 = 4
Solution: x = 4, y = 2
Method 2: Elimination
2x + y = 10 … (1) x - y = 2 … (2)
Add (1) and (2): 3x = 12 x = 4
Substitute in (2): 4 - y = 2 y = 2
🔢 Quadratic Equations
Standard Form
ax² + bx + c = 0
Where a ≠ 0
Methods to Solve
Method 1: Factorization
Example: x² - 5x + 6 = 0
Find two numbers that:
- Multiply to give c (6)
- Add to give b (-5)
Numbers: -2 and -3 (x - 2)(x - 3) = 0
x = 2 or x = 3
Method 2: Quadratic Formula
x = [-b ± √(b² - 4ac)] / 2a
Example: x² - 5x + 6 = 0 a = 1, b = -5, c = 6
x = [5 ± √(25 - 24)] / 2 x = [5 ± 1] / 2 x = 6/2 or 4/2 x = 3 or 2
Method 3: Completing the Square
x² - 5x + 6 = 0 x² - 5x = -6 x² - 5x + (5/2)² = -6 + (5/2)² (x - 5/2)² = 1/4 x - 5/2 = ±1/2 x = 3 or 2
💡 Solved Examples - Equations
Example 1: Simple Linear
Q: Solve: 4x - 7 = 13
Solution:
4x = 13 + 7 4x = 20 x = 5
Answer: x = 5
Example 2: Two Variables
Q: Solve:
3x + 2y = 16 2x - y = 3
Solution:
From equation 2: y = 2x - 3
Substitute in equation 1: 3x + 2(2x - 3) = 16 3x + 4x - 6 = 16 7x = 22 x = 22/7
y = 2(22/7) - 3 = 44/7 - 21/7 = 23/7
Answer: x = 22/7, y = 23/7
Example 3: Quadratic
Q: Solve: x² - 7x + 12 = 0
Solution:
Factorize: Numbers that multiply to 12 and add to -7: -3, -4
(x - 3)(x - 4) = 0 x = 3 or x = 4
Answer: x = 3, 4
📊 Inequalities (Inequations)
Basic Rules
Rule 1: Adding/Subtracting Same Number
If a > b, then: a + c > b + c a - c > b - c
Sign doesn’t change!
Rule 2: Multiplying/Dividing by Positive
If a > b and c > 0, then: ac > bc a/c > b/c
Sign doesn’t change!
Rule 3: Multiplying/Dividing by Negative
If a > b and c < 0, then: ac < bc (sign reverses!) a/c < b/c (sign reverses!)
IMPORTANT: Sign flips when multiplying/dividing by negative!
Solving Inequalities
Example 1:
2x + 5 > 13
2x > 13 - 5 2x > 8 x > 4
Example 2:
-3x + 6 ≤ 15
-3x ≤ 15 - 6 -3x ≤ 9 x ≥ -3 (sign flips because dividing by -3)
🎯 IBPS Pattern: Comparing Two Quadratics
Common Question Format:
Given two quadratic equations, compare roots:
I. x² - 5x + 6 = 0 II. y² - 7y + 12 = 0
Find relationship between x and y.
Solution Steps:
Step 1: Solve both equations
I. x² - 5x + 6 = 0 (x-2)(x-3) = 0 x = 2 or 3
II. y² - 7y + 12 = 0 (y-3)(y-4) = 0 y = 3 or 4
Step 2: Compare all possible pairs
If x = 2: y can be 3 or 4 2 < 3 ✓, 2 < 4 ✓
If x = 3: y can be 3 or 4 3 = 3 ✓, 3 < 4 ✓
Conclusion: x ≤ y
Options Usually:
A) x > y B) x ≥ y C) x < y D) x ≤ y E) x = y or no relation
Answer: D (x ≤ y)
💡 More Examples
Example 4: Inequality Solving
Q: Solve: 3(x - 2) < 2(x + 1)
Solution:
3x - 6 < 2x + 2 3x - 2x < 2 + 6 x < 8
Answer: x < 8
Example 5: Compound Inequality
Q: Solve: 5 < 2x + 1 < 13
Solution:
Split into two: 5 < 2x + 1 AND 2x + 1 < 13
From first: 4 < 2x 2 < x
From second: 2x < 12 x < 6
Combined: 2 < x < 6
Answer: 2 < x < 6
Example 6: IBPS Type Comparison
Q:
I. x² = 25 II. y² - 11y + 30 = 0
Solution:
I. x² = 25 x = ±5 x = 5 or -5
II. y² - 11y + 30 = 0 (y - 5)(y - 6) = 0 y = 5 or 6
Compare: If x = 5: y = 5 (x = y) or y = 6 (x < y) If x = -5: y = 5 (-5 < 5) or y = 6 (-5 < 6)
No definite relation (sometimes equal, sometimes x < y)
Answer: E (No relation or x = y)
⚡ Quick Shortcuts
Shortcut 1: Sum and Product of Roots
For ax² + bx + c = 0:
Sum of roots = -b/a Product of roots = c/a
Example: x² - 5x + 6 = 0 Sum = 5, Product = 6 Roots are 2 and 3 ✓
Shortcut 2: Perfect Square Check
For x² + 2px + p² = 0: This is (x + p)² = 0 Both roots are -p (equal roots)
Shortcut 3: Factorization Pattern
x² + (a+b)x + ab = (x+a)(x+b)
Example: x² + 7x + 12 Find a, b: a×b = 12, a+b = 7 a = 3, b = 4 (x+3)(x+4) = 0
Shortcut 4: Inequality Comparison
For x² - ax + b = 0: If both roots positive: sum > 0, product > 0 If both roots negative: sum < 0, product > 0 If roots have opposite signs: product < 0
⚠️ Common Mistakes
❌ Mistake 1: Sign Flip Forgotten
Wrong: -2x > 6 → x > -3 ✗ Right: -2x > 6 → x < -3 ✓ (Sign flips when dividing by negative!)
❌ Mistake 2: Missing Negative Root
Wrong: x² = 16 → x = 4 only ✗ Right: x² = 16 → x = ±4 ✓
❌ Mistake 3: Incomplete Comparison
Wrong: Comparing only one pair of roots ✗ Right: Compare ALL possible combinations ✓
❌ Mistake 4: Calculation Error in Factorization
Always verify: (x-a)(x-b) = x² - (a+b)x + ab
📊 IBPS Relationship Codes
Standard Answer Options:
A) x > y (x is always greater) B) x ≥ y (x is greater than or equal) C) x < y (x is always smaller) D) x ≤ y (x is smaller than or equal) E) x = y or no relation can be established
How to Choose:
- If ALL comparisons show same relation → Choose A, B, C, or D
- If relations are mixed → Choose E
📝 Practice Problems
Level 1:
- Solve: 5x - 3 = 17
- Solve: x² - 9 = 0
- Solve: 2x + 5 < 15
Level 2:
- Solve: 3x + 2y = 12, x - y = 1
- Solve: x² - 6x + 8 = 0
- Solve: -4x + 8 ≥ 20
Level 3:
- Compare: I. x² - 8x + 15 = 0, II. y² - 6y + 8 = 0
- Solve: x² + x - 12 = 0
- Solve: 3 ≤ 2x - 1 ≤ 11
🔗 Related Topics
Prerequisites:
- Number System - Basic operations
- Simplification - Solving equations
Related:
- Data Sufficiency - Uses equations
Practice:
🎯 Continue Your Learning Journey
Master Equations - Practice quadratic comparisons daily! ⚖️
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