Linear Inequations
Introduction
A linear inequation is a mathematical statement that relates two expressions using inequality symbols (<, >, ≤, ≥). In ICSE Class 10, you learn to solve linear inequations in one variable and represent the solution set on a number line.
Inequality Symbols
| Symbol | Meaning | Example |
|---|---|---|
| < | Less than | x < 3 |
| > | Greater than | x > 5 |
| ≤ | Less than or equal to | x ≤ 2 |
| ≥ | Greater than or equal to | x ≥ 4 |
Replacement Set and Solution Set
- Replacement set — The set of numbers from which values can be chosen (often natural numbers N, whole numbers W, integers I/Z, or real numbers R).
- Solution set — The subset of the replacement set that satisfies the inequation.
Rules for Solving Inequations
Rule 1: Adding or Subtracting
Adding (or subtracting) the same number to both sides does NOT change the inequality.
If a ≤ b, then a + c ≤ b + c
Rule 2: Multiplying or Dividing by a Positive Number
Multiplying (or dividing) both sides by a positive number does NOT change the inequality.
If a ≤ b and c > 0, then ac ≤ bc
Rule 3: Multiplying or Dividing by a Negative Number
Multiplying (or dividing) both sides by a negative number REVERSES the inequality.
If a ≤ b and c < 0, then ac ≥ bc
This is the most common source of errors — never forget to reverse the sign!
Worked Examples
Example 1: Basic Inequation (Replacement Set = Integers)
Solve 2x + 3 ≤ 15, where x ∈ I (integers). Represent the solution set on a number line.
Solution: 2x + 3 ≤ 15 2x ≤ 15 − 3 2x ≤ 12 x ≤ 6
Since x ∈ I (integers), solution set = {..., −2, −1, 0, 1, 2, 3, 4, 5, 6}
Number line representation: Shade all points from the left up to and including 6.
Example 2: Negative Coefficient
Solve 5 − 3x > 11, x ∈ W (whole numbers).
Solution: 5 − 3x > 11 −3x > 11 − 5 −3x > 6
Divide by −3 (reverse the sign): x < −2
Since x ∈ W (whole numbers = {0, 1, 2, ...}), no whole number satisfies x < −2. Solution set = ∅ (empty set)
Example 3: Compound Inequation
Solve −3 ≤ 2x − 1 < 5, x ∈ R. Represent on a number line.
Solution: Solve the two parts together: −3 ≤ 2x − 1 AND 2x − 1 < 5
Part 1: −3 ≤ 2x − 1 −3 + 1 ≤ 2x −2 ≤ 2x −1 ≤ x or x ≥ −1
Part 2: 2x − 1 < 5 2x < 6 x < 3
Combining: −1 ≤ x < 3
Solution set = {x ∈ R: −1 ≤ x < 3}
Example 4: Word Problem
The sum of three consecutive odd natural numbers is at most 60. Find the largest possible set of such numbers.
Solution: Let the numbers be x, x + 2, x + 4 (consecutive odd numbers). Sum = x + (x + 2) + (x + 4) = 3x + 6 ≤ 60 3x ≤ 54 x ≤ 18
Since numbers are odd natural numbers: x can be 1, 3, 5, 7, 9, 11, 13, 15, 17 The largest set is 17, 19, 21.
Largest possible numbers: 17, 19, 21
Comparison: Open vs Closed Intervals
| Interval Notation | Inequality | Number Line |
|---|---|---|
| (a, b) | a < x < b | Open circles at a and b |
| [a, b] | a ≤ x ≤ b | Closed circles at a and b |
| [a, b) | a ≤ x < b | Closed at a, open at b |
| (a, ∞) | x > a | Open circle at a, arrow to right |
| (−∞, b] | x ≤ b | Closed circle at b, arrow to left |
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| Not reversing inequality when multiplying/dividing by negative | Always reverse the sign when multiplying/dividing by a negative number |
| Mixing up open and closed circles on number line | Strict inequality (< or >) = open circle; ≤ or ≥ = closed circle |
| Including incorrect numbers from replacement set | Always check if the replacement set includes the boundary value |
| Writing solution set without checking domain | Verify each value against the original inequation |
ICSE Exam Focus
Linear inequations typically carry 6–8 marks in ICSE exams. Questions commonly require:
- Solving a linear inequation with integer/real replacement set.
- Representing the solution on a number line.
- Solving compound inequations.
- Word problems involving inequations.
Marks Blueprint:
| Topic | Marks |
|---|---|
| Simple inequation and solution set | 3 |
| Number line representation | 2 |
| Compound inequation | 3 |
| Word problem | 2–3 |
Self-Test Questions
-
Solve 4x − 7 > 13, x ∈ N. Represent the solution set on a number line.
-
Solve −5 ≤ 3x + 4 < 16, x ∈ I (integers). Write the solution set.
-
The length of a rectangle is 5 cm more than its width. If the perimeter is at most 50 cm, find the possible widths.
-
Solve (2x − 3)/4 < (x + 1)/2, x ∈ R. Represent on a number line.
-
Explain the rule for multiplying an inequation by a negative number. Why does the inequality sign reverse?
