NCERT Class 11 Linear Inequalities: Concepts, Graphs, and Solutions

Imagine deciding how much money you can spend without exceeding your budget, or figuring out the minimum number of products to sell to make a profit. These situations don’t demand exact values—they allow a range of possibilities. This is where linear inequalities come in. They represent conditions with flexibility, unlike strict equations. In Class 11 Maths, this chapter is crucial not just for exams but also for real-life problem solving, helping you think logically about limits, ranges, and constraints.

What are Linear Inequalities? 

An inequality is a mathematical statement that compares two expressions using symbols such as greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). Unlike equations, which show equality between two expressions, inequalities describe a relationship where one side is larger or smaller than the other.

For example:

• 5 > 3 shows that 5 is greater than 3
• x ≤ 4 means x can be any value less than or equal to 4
• 2x + 3 > 7 represents a condition involving a variable 

The key difference between an equation and an inequality is that equations have a fixed solution (or set of exact solutions), while inequalities usually have a range of solutions.

Types based on variables:

1. Numerical Inequalities
These do not involve variables.
Example: 3 < 8, 10 ≥ 5

2. Literal Inequalities
These involve variables.
Example: x > 3, y ≤ 5

3. Linear Inequalities
A linear inequality involves variables with the highest power as 1.

• In one variable: 3x – 2 < 0
• In two variables: 2x + 3y ≥ 4 

Linear inequalities in one variable are usually represented on a number line, while those in two variables are represented on a graph as a region.

Types of Inequalities

Inequalities can be broadly classified into two main types based on the symbols used:

1. Strict Inequalities

These use the symbols greater than (>) or less than (<).
They do not include the boundary value.

Examples:

• x > 5 (x is strictly greater than 5)
• 2x – 1 < 3 

In these cases, the value where equality occurs is not included in the solution.

2. Non-Strict (Slack) Inequalities

These use the symbols greater than or equal to (≥) or less than or equal to (≤).
They include the boundary value.

Examples:

• x ≥ 5 (includes 5 and all values greater than 5)
• y ≤ 10 

Understanding Inclusion and Exclusion

This concept is very important:

• If the inequality has ≥ or ≤, the boundary value is included
• If the inequality has > or <, the boundary value is excluded 

For example:

• x ≥ 3 includes 3
• x > 3 does not include 3 

This distinction becomes especially important when representing solutions on number lines or graphs.

Solution of an Inequality (Core Concept)

The solution of an inequality is the set of all values of the variable that make the inequality true. Unlike equations, inequalities usually have infinitely many solutions.

For example:

• x – 1 ≥ 0 → x ≥ 1
  All real numbers greater than or equal to 1 satisfy this inequality. 

There are also cases where no solution exists.
For example:

• x² + 1 < 0
  Since x² is always non-negative, x² + 1 is always positive. So, no real value of x satisfies this inequality. 

Rules for Solving Inequalities

These rules are essential:

1. Add or subtract the same number on both sides
The inequality sign does not change.

2. Multiply or divide by a positive number
The inequality sign remains the same.

3. Multiply or divide by a negative number
The inequality sign reverses.

Example 1: Solve 3x – 5 < x + 7

Step 1: Bring like terms together
3x – x < 7 + 5
2x < 12

Step 2: Divide by 2
x < 6

So, all values less than 6 are solutions.

Example 2: Solve –2x ≥ 4

Step 1: Divide both sides by –2
x ≤ –2

Note: The inequality sign reverses because we divided by a negative number.

Representation on Number Line

Solutions of linear inequalities in one variable can be represented on a number line. This helps in visualizing the range of solutions.

Key Concepts

1. Open Circle
Used when the inequality is strict (>, <).
It shows that the boundary value is not included.

2. Closed Circle
Used when the inequality is non-strict (≥, ≤).
It shows that the boundary value is included.

Direction of Shading

• For x < a → shade towards the left
• For x > a → shade towards the right 

Examples

1. x < 3
Draw an open circle at 3 and shade all values to the left.

2. x ≥ –2
Draw a closed circle at –2 and shade all values to the right.

This method makes it easier to understand which values satisfy the inequality and helps avoid mistakes in exams.

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Graphical Representation in Two Variables

A linear inequality in two variables is generally written in the form ax + by > c, ax + by ≥ c, ax + by < c, or ax + by ≤ c, where a and b are not both zero. Unlike one-variable inequalities, their solutions are not points but entire regions on a graph.

Steps to Represent Graphically

Step 1: Convert the inequality into an equation
Replace the inequality sign with an equality sign.
Example: 2x + 3y > 6 becomes 2x + 3y = 6

Step 2: Draw the line
Plot the straight line corresponding to the equation by finding two points.

Step 3: Decide the correct region using a test point
Choose a point not on the line, usually (0, 0).
Substitute it into the inequality:

• If it satisfies the inequality, shade that side
• If not, shade the opposite side 

Dotted vs Solid Line

• Use a dotted line for strict inequalities (>, <) because points on the line are not included
• Use a solid line for non-strict inequalities (≥, ≤) because points on the line are included 

Half-Plane Concept

The graph of a linear inequality divides the plane into two regions called half-planes. One of these regions satisfies the inequality, and that shaded region represents the solution set.

Important Results and Properties

Understanding certain properties helps simplify inequalities quickly and avoid mistakes.

1. Product Sign Rules

These rules are useful when dealing with expressions involving multiplication or fractions:

• If ab > 0, then a and b have the same sign (both positive or both negative)
• If ab < 0, then a and b have opposite signs 

These rules are especially helpful when solving inequalities involving rational expressions.

2. Modulus (Absolute Value) Rules

Absolute value represents distance from zero, so it always gives a non-negative value.

• If |x| < a, then –a < x < a
• If |x| ≤ a, then –a ≤ x ≤ a
• If |x| > a, then x < –a or x > a
• If |x| ≥ a, then x ≤ –a or x ≥ a 

These rules convert modulus inequalities into simpler linear inequalities that can be solved easily.

Solved Examples (Concept Building)

Example 1: Basic Inequality

Solve: 3x – 5 < x + 7

Step 1: Subtract x from both sides
3x – x – 5 < 7
2x – 5 < 7

Step 2: Add 5
2x < 12

Step 3: Divide by 2
x < 6

Final Answer: x < 6

Example 2: Fraction Inequality

Solve: (x + 2)/(x + 5) > 0

The expression is positive when numerator and denominator have the same sign.

Case 1: x + 2 > 0 and x + 5 > 0 → x > –2
Case 2: x + 2 < 0 and x + 5 < 0 → x < –5

Final Answer: x < –5 or x > –2

Example 3: Modulus Inequality

Solve: |x – 2| ≤ 3

–3 ≤ x – 2 ≤ 3

Add 2 to all parts:
–1 ≤ x ≤ 5

Final Answer: –1 ≤ x ≤ 5

Example 4: Double Inequality

Solve: 1 ≤ 2x + 3 ≤ 7

Subtract 3:
–2 ≤ 2x ≤ 4

Divide by 2:
–1 ≤ x ≤ 2

Final Answer: –1 ≤ x ≤ 2

Example 5: Real-Life Application (Profit Problem)

Cost function: C(x) = 20x + 4000
Revenue function: R(x) = 60x + 2000

Profit = R(x) – C(x)
= (60x + 2000) – (20x + 4000)
= 40x – 2000

For profit:
40x – 2000 > 0
40x > 2000
x > 50

Final Answer: More than 50 items must be sold

Example 6: Case-Based Inequality

Solve: |x + 1| + |x| > 3

Critical points: x = –1 and x = 0

Case 1: x < –1
–(x + 1) – x > 3
–2x – 1 > 3
–2x > 4
x < –2

Case 2: –1 ≤ x < 0
(x + 1) – x > 3
1 > 3 (not possible)

Case 3: x ≥ 0
(x + 1) + x > 3
2x + 1 > 3
2x > 2
x > 1

Final Answer: x < –2 or x > 1

Special Case: Absolute Value Inequalities

Absolute value represents the distance of a number from zero on the number line. That is why it is always non-negative.

For example, |x – 2| represents the distance of x from 2.

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Case Splitting Method

To solve modulus inequalities, we split them into cases based on definition:

• If |x| = a, then x = a or x = –a
• If |x| < a, convert into a double inequality
• If |x| > a, split into two separate inequalities 

Examples

1. Solve: |x – 2| ≤ 3
–3 ≤ x – 2 ≤ 3
–1 ≤ x ≤ 5

2. Solve: |x| > 5
x < –5 or x > 5

This method helps simplify complex inequalities into basic linear forms.

Also Read

Binomial Theorem Made Easy for Class 11 Students

System of Linear Inequalities

A system of linear inequalities consists of two or more inequalities taken together. The solution is the set of values that satisfy all inequalities simultaneously.

Key Concepts

• Each inequality represents a region
• The common region satisfying all inequalities is the solution
• This region is found by overlapping shaded areas 

Graphical Meaning

When graphed, each inequality divides the plane into two half-planes. The intersection of these half-planes gives the final solution region.

No Solution Case

If there is no overlapping region, the system has no solution.

Example

Solve:
x ≥ 2
x < 5

The common solution is:
2 ≤ x < 5

This means all values between 2 and 5, including 2 but not 5.

This concept is very important for higher mathematics, especially in optimization and real-life decision-making problems.

Application-Based Problems (Real-Life Use)

Linear inequalities are not just theoretical concepts—they are widely used in real-life situations where limits and ranges matter.

1. Profit and Loss
Businesses use inequalities to determine minimum sales required to make a profit. For example, if profit must be greater than zero, it forms an inequality that helps find the required number of units to sell.

2. Temperature Range
In science and daily life, certain conditions must remain within a range. For instance, maintaining a temperature between 20°C and 30°C can be expressed using inequalities.

3. Mixture Problems
When mixing solutions of different concentrations, inequalities help determine the quantity needed to achieve a desired concentration range.

4. Geometry-Based Problems
Inequalities are used to define constraints like perimeter, area, or side lengths. For example, ensuring the perimeter of a shape exceeds a certain value leads to an inequality.

These applications show how inequalities help in decision-making, planning, and maintaining conditions within limits.

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Common Mistakes Students Make

Students often lose marks in this chapter due to small but critical errors.

• Forgetting to reverse the sign
  When multiplying or dividing by a negative number, the inequality sign must be reversed. This is one of the most common mistakes.
• Wrong interval notation
  Students sometimes confuse open and closed intervals, especially when dealing with strict and non-strict inequalities.
• Graph confusion
  Using a solid line instead of a dotted line (or vice versa) can lead to incorrect graphical representation.
• Ignoring the domain
  If the question specifies natural numbers, integers, or real numbers, the solution must be written accordingly. Ignoring this can make an otherwise correct answer incomplete. 

Avoiding these mistakes can significantly improve accuracy in exams.

13. Practice Set (Category-Wise)

A. Basic Inequalities

1. Solve: 2x + 3 < 11
2. Solve: 5x – 7 ≥ 8
3. Solve: 3 – x ≤ 6
4. Solve: 4x + 1 > 9
5. Solve: 7x – 2 ≤ 5x + 6 

B. Moderate Level

6. Solve: 3x – 5 < 2x + 7
7. Solve: (x + 4)/2 ≥ 3
8. Solve: 2 – (3x/4) > 1
9. Solve: 5x + 2 ≤ 3x + 10
10. Solve: (2x – 1)/3 < (x + 2)/2 

C. Modulus-Based

11. Solve: |x| < 4
12. Solve: |x – 3| ≤ 2
13. Solve: |2x + 1| > 5
14. Solve: |x + 2| ≥ 6
15. Solve: |3x – 4| < 8 

D. Case-Based / Word Problems

16. The cost of producing x items is 50x + 1000. If revenue is 80x, find the minimum x for profit.
17. A classroom temperature must stay between 18°C and 25°C. Represent this using inequalities.
18. A solution must contain more than 10% but less than 20% salt. Form an inequality.
19. The perimeter of a rectangle is more than 100 cm. If length = 2x and breadth = x, find x.
20. A person must score at least 40 marks but not more than 90 marks. Represent this range. 

E. MCQs

21. If x > 3, then –x is:
    (a) > –3 (b) < –3 (c) ≥ –3 (d) ≤ –3
22. Solution of |x| < 2 is:
    (a) x > 2 (b) x < –2 (c) –2 < x < 2 (d) x ≥ 2
23. If 2x – 5 ≥ 1, then x is:
    (a) ≥ 3 (b) ≤ 3 (c) > 3 (d) < 3
24. Which is a strict inequality?
    (a) x ≥ 5 (b) x ≤ 3 (c) x > 2 (d) x = 4
25. If x ≤ 4, the interval is:
    (a) (–∞, 4) (b) (–∞, 4] (c) [–∞, 4] (d) (–∞, 4] 

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F. Assertion-Reason / Conceptual

26. Assertion: Multiplying both sides of an inequality by a negative number reverses the sign.
    Reason: Negative numbers change the direction of inequality.
27. Assertion: |x| is always positive.
    Reason: Absolute value represents distance from zero.
28. Assertion: x > 5 includes 5.
    Reason: Strict inequalities include boundary values.
29. Assertion: Graph of x ≥ 2 includes the point 2.
    Reason: Non-strict inequalities include boundary points.
30. Assertion: x² + 1 < 0 has real solutions.
    Reason: Square of a number is always non-negative. 

Why Learn Linear Inequalities with PlanetSpark

• Concept Clarity Over Rote Learning
  Students focus on understanding the logic behind inequalities instead of just memorizing formulas and steps.
• Interactive Learning Experience
  Concepts are taught through engaging sessions, making learning more interesting and easier to grasp.
• Real-Life Application Focus
  Students learn how inequalities are used in real-world scenarios like budgeting, business decisions, and measurements.
• Guided Problem-Solving Approach
  Step-by-step explanations help students develop strong analytical and problem-solving skills.
• Personal Mentorship
  Each student gets individual attention, ensuring doubts are cleared instantly and concepts are well understood.
• Confidence Building
  Regular practice and feedback help students become more confident in solving questions independently.
• Strong Foundation for Higher Mathematics
  A clear understanding of inequalities prepares students for advanced topics in algebra, calculus, and beyond.

Where Logic Meets Real-Life Decisions

Linear inequalities teach more than just solving problems—they build the ability to think within limits and make logical decisions. From understanding ranges to analyzing real-life constraints, this chapter plays an important role in both exams and future studies. With consistent practice and clear concepts, mastering inequalities becomes much easier. Focus on understanding the “why” behind each step, and you will not only score well but also develop strong mathematical thinking skills.

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