Introduction to Linear Inequations
A linear inequation in one variable x is an inequality of the form ax + b > 0, ax + b < 0, ax + b >= 0, or ax + b <= 0, where a, b are real numbers and a != 0.
Comparison with Linear Equations
While a linear equation has a unique (or fixed) solution, a linear inequation has a range of solutions.
Rules for Solving Inequations
Rule 1: Adding or subtracting the same number on both sides does not change the inequality.
If a > b, then a + c > b + c and a - c > b - c.
Rule 2: Multiplying or dividing by a positive number preserves the inequality.
If a > b and c > 0, then ac > bc and a/c > b/c.
Rule 3 (Critical): Multiplying or dividing by a negative number reverses the inequality.
If a > b and c < 0, then ac < bc and a/c < b/c.
Common Errors with Reversal
Students often forget to reverse the inequality sign when multiplying or dividing by a negative number. This is the most frequent mistake in solving inequations.
Solving Linear Inequations in One Variable
Example 1: Solve 3x - 5 < 7, x in R.
3x < 12
x < 4
Solution set = {x : x in R, x < 4}
Example 2: Solve -2x + 1 <= 5, x in R.
-2x <= 4
x >= -2 (inequality reversed due to division by -2)
Solution set = {x : x in R, x >= -2}
Graphical Representation on Number Line
Solutions of linear inequations are represented on the number line using:
- Open circle (o) for
<or>(endpoint not included) - Closed circle (.) for
<=or>=(endpoint included) - Dark line to show the solution region
Example: Represent x > 2 on a number line.
Place an open circle at 2 and darken the line to the right of 2.
System of Linear Inequations
Solve each inequation separately, then find the intersection of their solution sets.
Example: Solve 2x + 1 >= 5 and 3x - 2 < 10, x in R.
First: 2x >= 4, x >= 2
Second: 3x < 12, x < 4
Combined solution: 2 <= x < 4, i.e., x in [2, 4).
Interval Notation
| Inequality | Interval | Type |
|---|---|---|
a < x < b | (a, b) | Open |
a <= x <= b | [a, b] | Closed |
a <= x < b | [a, b) | Half-open |
a < x <= b | (a, b] | Half-open |
x >= a | [a, infinity) | |
x > a | (a, infinity) | |
x <= a | (-infinity, a] | |
x < a | (-infinity, a) |
Word Problems Involving Inequations
Example: A student needs an average of at least 60 marks in five subjects to pass. If he scored 52, 58, 63, and 55 in four subjects, what minimum marks must he score in the fifth subject?
Let x be the fifth subject marks.
(52 + 58 + 63 + 55 + x)/5 >= 60
228 + x >= 300
x >= 72
The student must score at least 72 marks.
ISC Application Problems
ISC frequently includes practical problems involving inequations such as:
- Age constraints in puzzles
- Speed and distance with time limits
- Cost and profit threshold problems
- Mixture and concentration problems
Graphical Solution of Linear Inequations in Two Variables
For ax + by + c > 0 or < 0:
- Draw the line
ax + by + c = 0(dashed for strict inequality, solid for<=or>=). - Choose a test point not on the line (usually origin if not on line).
- Substitute in the inequality. If true, shade the half-plane containing the test point.
Worked Examples
Example 1: Solve (2x - 1)/3 >= (3x - 2)/5 - 2, x in R.
Multiply by 15: 5(2x - 1) >= 3(3x - 2) - 30
10x - 5 >= 9x - 6 - 30
10x - 5 >= 9x - 36
x >= -31
Solution set = [-31, infinity).
Example 2: Solve 4x + 3 < 6x + 7 and 2x + 1 > 3x - 4.
First: 4x - 6x < 7 - 3, -2x < 4, x > -2
Second: 2x - 3x > -4 - 1, -x > -5, x < 5
Solution: -2 < x < 5, i.e., x in (-2, 5).
Common Mistakes
- Sign reversal: Not reversing inequality when multiplying/dividing by a negative number.
- Open vs closed interval: Confusing
<with<=on number line representation. - Intersection vs union: Taking union instead of intersection when solving a system of inequations.
- Domain specification: Always specify
x in N,x in Z, orx in Ras appropriate.
ISC Exam Focus
- Theory (70%): Solving methods, interval representation, algebraic manipulation.
- Application (30%): Word problems, graphical representation on number line.
- Marks: Short answer (2-3 marks) for solving simple inequations, long answer (4-6 marks) for systems and word problems.
- Algebraic errors and sign reversal are the most common exam pitfalls.
Self-Test Questions
Q1: Solve 5x - 3 < 2x + 9, x in R.
Answer: 5x - 2x < 9 + 3, 3x < 12, x < 4. Solution set = (-infinity, 4).
Q2: Solve -3(x - 2) >= 2x - 1.
Answer: -3x + 6 >= 2x - 1, -3x - 2x >= -1 - 6, -5x >= -7, x <= 7/5. Solution = (-infinity, 7/5].
Q3: Solve the system: 2x + 1 > 5 and 3x - 2 < 13.
Answer: First: x > 2. Second: x < 5. Solution: 2 < x < 5, i.e., (2, 5).
Q4: A man is 4 times as old as his son. After 10 years, he will be at most 3 times as old. Find the maximum possible present age of the son.
Answer: Let son's age = x, father's age = 4x. 4x + 10 <= 3(x + 10), 4x + 10 <= 3x + 30, x <= 20. Maximum son's age = 20 years.
Q5: Solve (x - 1)/(x + 2) > 0.
Answer: Either x - 1 > 0 and x + 2 > 0 => x > 1. Or x - 1 < 0 and x + 2 < 0 => x < -2. Solution: (-infinity, -2) cup (1, infinity).
Q6: Represent the solution of -2 <= x < 5, x in R, on a number line.
Answer: Closed circle at -2, open circle at 5, line connecting them.
