| --- |
| source: ncert |
| topic: algebra |
| class_level: class_11 |
| chapter: ch6 |
| difficulty: intermediate |
| --- |
| |
| Class 11 | Ch6: Linear Inequalities |
| RULES FOR INEQUALITIES: |
| Adding/subtracting same quantity: inequality sign UNCHANGED. |
| Multiplying/dividing by POSITIVE number: sign UNCHANGED. |
| Multiplying/dividing by NEGATIVE number: sign REVERSED. |
| SOLVING LINEAR INEQUALITIES IN ONE VARIABLE: |
| Solve like equation but reverse sign when multiplying/dividing by negative. |
| 3x-2>2x+1 → x>3. Solution: (3,infinity). On number line: open circle at 3, arrow right. |
| GRAPHICAL METHOD FOR TWO VARIABLES: |
| ax+by<c: boundary line ax+by=c (dashed for < or >, solid for <= or >=). |
| Test point (usually origin) to determine which side. |
| SYSTEM OF INEQUALITIES: |
| Find region satisfying ALL inequalities simultaneously (feasible region). |
| SOLVED EXAMPLES: |
| Example 1: Solve 3x-2>2x+1. |
| 3x-2x>1+2 → x>3. Solution set: (3,∞). |
| Example 2: Solve -3<(4x-1)/5<3. |
| Multiply by 5: -15<4x-1<15 → -14<4x<16 → -7/2<x<4. |
| Solution: (-3.5,4). |
| Example 3: Solve system x+y<=10, x+3y<=15, x>=0, y>=0. |
| Draw lines x+y=10 and x+3y=15. Find intersection: subtract to get 2y=5, y=2.5, x=7.5. |
| Feasible region: bounded by vertices (0,0),(10,0),(7.5,2.5),(0,5). |
| Example 4: IQ score: 62+4x/2<72 where x=age in months over 100. |
| 124+4x<144 → 4x<20 → x<5. So age < 105 months. |
| COMMON MISTAKES: |
| MUST reverse inequality when multiplying/dividing by NEGATIVE number. |
| For absolute value: |x|<a means -a<x<a (two conditions combined). |
| Dashed line for strict inequality (<,>). Solid line for (<=,>=). |
