Inequalities Class 11 Maths | Rules, Graphs & Exam Weightage (JEE, KCET, Boards)

Introduction

In mathematics, equations give us exact answers, while inequalities describe a range of possible values. An inequality tells us not just one solution, but often infinitely many values that satisfy a condition.

Examples from real life:

Economics: Profit > Cost (to ensure business sustainability).
Physics: Speed ≤ 80 km/h (traffic law condition).
Academics: Marks ≥ 35 (passing criteria).
Daily life: Remaining time ≤ 2 hours (deadline condition).

Thus, inequalities are everywhere. In NCERT Class 11 Chapter 5, inequalities form the basis of more advanced concepts like quadratic inequalities, modulus-based problems, and optimization in linear programming.

For competitive exams like JEE, KCET, COMEDK, and Boards, inequalities are important because they test:

Your ability to compare values logically.
Your skill in manipulating algebraic expressions.
Your understanding of graphs and intervals.

Symbols of Inequalities

An inequality is a mathematical statement involving two expressions compared with one of the following symbols:

There are four main inequality symbols:

Less than (<): x < y means x is strictly smaller than y.
- Example: 3 < 8.
Greater than (>): x > y means x is strictly larger than y.
- Example: 9 > 2.
Less than or equal to (≤): x ≤ y means x is smaller or equal to y.
- Example: Marks ≤ 100.
Greater than or equal to (≥): x ≥ y means x is larger or equal to y.
- Example: Age ≥ 18.

Strict inequalities (<, >) do not include boundary values.
Non-strict inequalities (≤, ≥) include boundary values.

Types of Inequalities

Linear Inequalities: Expressions of the form ax + b < c.
Example: 2x + 3 < 7.
Quadratic Inequalities: Involving x².
Example: x² − 4 ≥ 0.
Rational Inequalities: Involving fractions.
Example: 1/(x − 2) > 0.
Absolute Value Inequalities: Involving |x|.
Example: |x − 3| ≤ 2.

In Class 11, the main focus is on linear inequalities.

Rules for Manipulating Inequalities

Just like equations, inequalities can be manipulated. However, care must be taken because some operations reverse the inequality.

Rule 1: Addition and Subtraction

If a < b, then:

a + c < b + c
a − c < b − c

Adding or subtracting the same number preserves the inequality.

Example: If 5 < 8, then 5 + 3 < 8 + 3 → 8 < 11.

Rule 2: Multiplication and Division by a Positive Number

If a < b and c > 0, then:

a × c < b × c
a ÷ c < b ÷ c

Multiplying/dividing by a positive number does not change the inequality.

Example: If 2 < 5 and we multiply by 3 → 6 < 15.

Rule 3: Multiplication and Division by a Negative Number

If a < b and c < 0, then:

a × c > b × c
a ÷ c > b ÷ c

Multiplying/dividing by a negative number reverses the inequality.

Example: If 3 < 7, multiplying by -2 gives -6 > -14.

Rule 4: Transitive Property

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

Example: If 4 < 8 and 8 < 12, then 4 < 12.

Rule 5: Reciprocal Rule

If a < b and both are positive, then 1/a > 1/b.

Example: Since 2 < 5, reciprocals give 1/2 > 1/5.

⚠️ Works only when numbers are positive.

Rule 6: Squaring Inequalities

If both a and b are non-negative and a < b → a² < b².
If a < b and both are negative, then a² > b².

Example: -5 < -2 → (-5)² = 25 > 4 = (-2)².

Solving Inequalities: Step-by-Step

Example 1:

Solve 2x + 3 < 7.

Step 1: Subtract 3 → 2x < 4.
Step 2: Divide by 2 → x < 2.

Solution: All real numbers less than 2.

Example 2:

Solve -3x ≥ 6.

Step 1: Divide both sides by -3.

Reverse the inequality: x ≤ -2.

Solution: All real numbers less than or equal to -2.

Example 3:

Solve 1/(x − 2) > 0.

Step 1: Denominator positive → x − 2 > 0 → x > 2.

Solution: All real numbers greater than 2.

Graphical Representation of Inequalities

On a Number Line:

x > 3: Open circle at 3, shade right.
x ≥ 3: Closed circle at 3, shade right.
x < 0: Open circle at 0, shade left.
x ≤ 0: Closed circle at 0, shade left.

Examples:

Inequality: x ≤ 4 → solution is the interval (−∞, 4].
Inequality: −2 < x < 5 → solution is the interval (−2, 5).

Inequalities in Competitive Exams

JEE Main

Focus on algebraic manipulations and range problems.
Often linked with quadratic inequalities (next section).
Example: Solve |x − 2| < 3.

KCET / COMEDK

Usually direct 1-mark MCQs.
Example: Solve for x: 2x + 5 > 11.

Board Exams

2–4 marks short questions.
Expect “solve and represent graphically” type questions.

Common Mistakes to Avoid

Forgetting to reverse inequality when dividing by a negative.
Incorrectly squaring both sides without checking sign conditions.
Missing boundary values in ≤ or ≥ cases.
Not representing open/closed circles properly in graphs.

Advanced Example Problems

Example 1 (Board level):

Solve 5x − 7 ≤ 3x + 1.

→ 5x − 3x ≤ 1 + 7
→ 2x ≤ 8
→ x ≤ 4.

Graph: Closed circle at 4, shaded left.

Example 2 (JEE level):

Solve (x − 2)(x − 5) ≤ 0.

Critical points: x = 2, x = 5.
Sign analysis:

For x < 2 → product positive.
For 2 ≤ x ≤ 5 → product ≤ 0.
For x > 5 → product positive.

Solution: 2 ≤ x ≤ 5.

Example 3 (KCET level):

Solve 2x + 5 > 11.

→ 2x > 6 → x > 3.

Graph: Open circle at 3, shaded right.

FAQs

Q1: What does x > 5 mean?

It means all real numbers greater than 5, not including 5 itself.

Q2: Why do we flip inequality signs when dividing by a negative number?

Because multiplying or dividing by a negative reverses order. Example: 2 < 5, but multiplying by -1 → -2 > -5.

Q3: What is the solution set of x ≥ 0?

All real numbers greater than or equal to zero (includes 0).

Q4: Do inequalities always have infinite solutions?

Usually yes (ranges), but sometimes inequalities reduce to contradictions (no solution) or tautologies (all real numbers).

Q5: Are inequalities important for JEE?

Yes. They form the base for quadratic inequalities, modulus problems, and functions domain questions.

Practice Problems

Solve 7x − 4 < 10.
Solve −2x + 3 ≥ 7.
Solve (x − 1)(x − 3) > 0.
Represent x ≤ −2 on the number line.
Solve |x − 5| < 2.

Summary & Key Takeaways

Inequalities describe ranges of values rather than exact solutions.
Four main symbols: <, >, ≤, ≥.
Rules: Add/subtract → no change, Multiply/divide by positive → no change, Multiply/divide by negative → reverse inequality.
Graphical representation is important for clarity.
Applications in JEE/KCET/Board exams range from simple linear problems to advanced quadratic and modulus-based inequalities.