Learning Objectives
- Understand inequalities and their properties
- Solve linear inequalities in one variable algebraically
- Represent solutions on the number line and graphically
- Solve systems of linear inequalities in two variables
Key Concepts
Properties of Inequalities
For real numbers a, b, c: (1) If a > b then a ± c > b ± c. (2) If a > b and c > 0 then ac > bc (inequality preserved). (3) If a > b and c < 0 then ac < bc (inequality reversed when multiplying/dividing by negative number). This is the most crucial rule. (4) If a > b > 0 then 1/a < 1/b (reciprocal reverses inequality for positive numbers).
Linear Inequalities in One Variable
Form: ax + b > 0 (or ≥, <, ≤). Solve like linear equations, but reverse inequality sign when multiplying/dividing by a negative number. Solution is an interval on the number line. Notation: (a, b) = open interval; [a, b] = closed interval; (a, ∞) or (-∞, b] = unbounded intervals.
System of inequalities: Find solution sets of individual inequalities, then take intersection. Example: Solve 3x - 7 > 2 AND 5x + 3 < 23. Solution: x > 3 AND x < 4, so x ∈ (3, 4).
Linear Inequalities in Two Variables
Form: ax + by + c ≥ 0. Graph the line ax + by + c = 0, then determine which half-plane satisfies the inequality by testing a point (usually origin). Shaded region represents the solution. Solid line for ≥ or ≤ (boundary included); dashed line for > or < (boundary excluded). Solution of system of inequalities is the intersection (common region) of all individual solution regions.
Modulus Inequalities (JEE Important)
|x| < a ⟹ -a < x < a (a > 0). |x| > a ⟹ x < -a or x > a. |x - c| < a ⟹ c - a < x < c + a. These are frequently tested in JEE.
Summary
Linear inequalities follow similar rules to equations except that multiplying or dividing by a negative number reverses the inequality sign. Solutions of one-variable inequalities are intervals on the number line. Two-variable inequalities are represented by half-planes on the coordinate plane.
Important Terms
- Inequality: Mathematical statement comparing two expressions
- Solution set: Set of all values satisfying the inequality
- Half-plane: Region on one side of a line in the coordinate plane
- Interval notation: (a,b) open, [a,b] closed, [a,b) half-open
- System of inequalities: Multiple inequalities solved simultaneously
Quick Revision
- Multiplying/dividing by negative number reverses inequality sign
- |x| < a ⟹ -a < x < a; |x| > a ⟹ x < -a or x > a
- Graph line first, then test a point to determine half-plane
- System solution = intersection of all individual solutions
- Use open circle (hollow dot) for strict inequality, filled dot for ≤ or ≥
- 1/a < 1/b when a > b > 0 (reciprocal reverses for positives)