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 (<=,>=).