5. Linear Inequalities – Class 11 Mathematics

Introduction

In mathematics, equations give us precise solutions, but inequalities broaden the picture by allowing ranges of possible solutions. For instance:

• The equation x + 3 = 7 gives a single solution, x = 4.
• But the inequality x + 3 < 7 gives all real values of x < 4.

Inequalities are essential in everyday life: comparing quantities, deciding limits, or analyzing optimization problems. In mathematics, they form the base of linear programming (Class 12), play a crucial role in functions and domains, and appear in competitive exams (JEE, KCET, COMEDK) in both direct and indirect forms.

This chapter focuses on linear inequalities – inequalities where the highest power of the variable is 1. It is divided into two key subtopics:

1. 5.2 Inequalities – introduction, rules, and properties.
2. 5.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation – solving step-by-step and representing results visually.

5.2 Inequalities

Definition

An inequality is a mathematical statement that compares two values or expressions using symbols:

• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)

Example:

• 5 < 8 (true statement).
• x + 2 ≥ 7 means all values of x for which x + 2 is at least 7.

Types of Inequalities

1. Strict inequalities – use < or >.
   Example: x > 3.
2. Non-strict inequalities – use ≤ or ≥.
   Example: y ≤ 5.

Rules of Inequalities

To solve inequalities, we use rules similar to equations, with one important difference: when multiplying or dividing by a negative number, the inequality sign reverses.

1. Addition/Subtraction Rule

If a < b, then:

• a + c < b + c
• a – c < b – c

Example: If x < 5, then x + 2 < 7.

2. Multiplication Rule

If a < b and c > 0, then ac < bc.
If a < b and c < 0, then ac > bc.

Example:

• If x < 4, then 2x < 8.
• If x < 4, then -2x > -8.

3. Division Rule

Same as multiplication: dividing by a negative reverses the inequality.

Example:
If 2x > 6, then dividing by 2 → x > 3.
If -2x > 6, then dividing by -2 (negative) → x < -3.

4. Transitive Property

If a < b and b < c, then a < c.

Example: If 2 < 5 and 5 < 9, then 2 < 9.

5. Squaring

If a, b ≥ 0 and a < b, then a² < b².

6. Reciprocals

If a, b > 0 and a < b, then 1/a > 1/b.

Examples

Example 1: Solve 3x + 4 < 10.
3x < 6 → x < 2.

Example 2: Solve -2x + 5 ≥ 1.
-2x ≥ -4. Dividing by -2 flips sign: x ≤ 2.

Exam Application (JEE/KCET/COMEDK)

• Direct inequalities (1-mark MCQs).
• Indirect: used in functions and domains (like ensuring square root ≥ 0 or denominator ≠ 0).
• Often linked with quadratic inequalities.

Marks Distribution

• Boards: 2–3 marks direct questions.
• JEE Main: 1 question (~4 marks, often via quadratic).
• KCET/COMEDK: 1 question (1 mark, speed-based).

5.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation

General Form

A linear inequality in one variable has the form:

ax + b < c or ax + b ≥ c

where a, b, c are real constants and a ≠ 0.

Steps to Solve

1. Simplify the inequality.
2. Isolate the variable.
3. Reverse sign if multiplying/dividing by negative.
4. Write a solution in interval form.
5. Represent the solution graphically.

Graphical Representation on Number Line

• x > a → open circle at a, shade right side.
• x ≥ a → closed circle at a, shade right side.
• x < a → open circle at a, shade left side.
• x ≤ a → closed circle at a, shade left side.

Example:
Solve 2x – 1 ≥ 3.
2x ≥ 4 → x ≥ 2.
Graph: closed circle at 2, shading rightwards.

Worked Examples

Example 1: Solve 5x – 7 < 3.
5x < 10 → x < 2.
Graph: open circle at 2, shading left.

Example 2: Solve 2 – 3x ≥ -4.
-3x ≥ -6 → dividing by -3 flips sign → x ≤ 2.

Example 3 (JEE type): Solve |2x – 1| < 3.
Two cases:

• 2x – 1 < 3 → x < 2.
• 2x – 1 > -3 → x > -1.
  Solution: -1 < x < 2.

Interval Notation

• x > 2 → (2, ∞).
• x ≥ 2 → [2, ∞).
• x < -1 → (-∞, -1).
• -1 < x < 2 → (-1, 2).

Applications in Exams

• Board exams: direct “solve and graph” problems.
• JEE: domain/range problems, quadratic inequalities.
• KCET/COMEDK: quick check inequalities in objective problems.

Practice Problems

1. Solve and represent: 3x + 2 > 8.
2. Solve: -2x + 5 ≤ 1.
3. Solve: |x – 4| ≥ 3.
4. Find the interval for which (2x – 1)/(x + 3) > 0.
5. Solve for x: 2x² – 5x + 2 ≥ 0.

FAQs

Q1. What’s the difference between linear equations and linear inequalities?
Equations have exact solutions; inequalities give ranges.

Q2. Why do we flip inequality sign when multiplying/dividing by a negative?
Because the direction of inequality changes when order is reversed.

Q3. Are inequalities important for JEE?
Yes — they appear indirectly in functions, domain restrictions, and quadratic questions.

Q4. How are solutions represented graphically?
On a number line with open/closed circles and shading.

Q5. What’s the mark’s weightage?

• Boards: ~3 marks.
• JEE: ~4 marks (1 problem).
• KCET/COMEDK: ~1 mark.

Summary and Key Takeaways

• Inequalities generalize equations by describing ranges instead of exact values.
• Rules are similar to equations, but flipping the inequality when multiplying/dividing by negative is crucial.
• Solutions are expressed in interval notation and represented graphically.
• This chapter builds foundation for Class 12 Linear Programming and higher algebra topics.
• Marks weightage: Boards (3–4), JEE (4), KCET/COMEDK (1).