Linear Inequalities Class 11

Linear Inequalities for class 11 notes are provided here. All the concepts in class 11 linear inequalities are covered here as per the CBSE syllabus, which helps students to score good marks in the examinations. The linear inequalities class 11 notes covers the various topics, such as inequalities, algebraic solutions for linear inequalities, graphical solutions for linear inequalities with many solved examples.

Linear Inequalities Class 11 Topics

The topics and subtopics covered in Linear inequalities for class 11 include:

  • Introduction
  • Inequalities
  • Algebraic Solutions for Linear Inequalities in One Variable and its Graphical Representation
  • Graphical Solution of Linear Inequalities in Two Variables

Introduction

In the earlier classes, we have studied the linear equations in one or two variables. In class 11, linear inequalities, we are going to learn about the linear inequalities in one variable, two variables with its algebraic solution and graphical solution. The linear inequalities are used to solve the problems in different fields like Science, Engineering, Mathematics, and so on.

Inequalities

Two algebraic expressions or real numbers related by the symbol ≤, ≥, < and > form an inequality. For example: px + qy > 0, 9a – 21b < 0, etc. Equal numbers can be subtracted or added from both the sides of an inequality equation. Also, both sides of an inequality can be divided or multiplied by the same number (non-zero). If both sides of an inequality are divided by the negative number, then the inequality equation gets reversed. The solution of the inequality is the value of x, which makes inequality a true statement.

Algebraic Solutions for Linear Inequalities in One Variable and its Graphical Representation

We can find the solution for the linear inequality using the trial and error method. But, sometimes, this method is not feasible, and it takes more time to compute the solution. So, we can solve the linear inequality using the numerical approach. Follow the below rules while solving the linear inequalities:

Rule 1: Add or subtract the same number on both the sides of an equation, without affecting the sign of the inequality

Rule 2: Multiply or divide both sides of an inequality equation by the same positive number.

Now, let us discuss a few examples of solving the linear inequalities in one variable and its graphical representation.

Example 1: 

Find the value of the variable “x” for the inequality 4x+3 < 6x+7

Solution:

Given inequality: 4x+3 < 6x+7

Now, add -6x on both sides of the inequality

4x+3 -6x < 6x+7-6x

Now, we get

⇒ -2x+3 < 7

⇒-2x < 4

⇒ x >-2

Therefore, all the real numbers which are greater than -2 are the solution for the given inequality.

Therefore, the solution set is (–2, ∞).

Example 2: 

Solve the linear inequality (3x-4)/2 ≥ (x+1)/4 -1, and represent the solution on the number line.

Solution:

Given: (3x-4)/2 ≥ (x+1)/4 -1

The given inequality is written as:

(3x-4)/2 ≥ (x-3)/4

2(3x-4)  ≥ (x-3)

6x-8 ≥ x-3

5x ≥ 5

X ≥ 1

Hence, the graphical representation for the inequality x ≥ 1 is given below:

Linear Inequality Example - 2

Graphical Solution of Linear Inequalities in Two Variables

We know that the line divides the cartesian plane into two parts, called the half-plane. The vertical line divides the plane into left and right half-planes, whereas the non-vertical line divides the plane into lower and upper half-planes. The region that contains all the solutions of an inequality is called the solution region. Now, let us solve the inequality graphically.

Example 3: 

Graphically solve the inequality  3x – 6 ≥ 0

Solution:

Given inequality is 3x – 6 ≥ 0

Now, substitute x=0 in the inequality, we get

3 (0) – 6 ≥ 0 

– 6 ≥ 0, which is false.

Hence, 3x – 6 ≥ 0 should be x ≥2

(i.e.,) 3x≥ 6

x ≥2

Hence, the inequality 3x – 6 ≥ 0 is graphically represented as:

Linear Inequality Example-3

Practice Problems

Solve the linear inequality class 11 problems given below:

  1. Solve the inequality 3 (2 – x) ≥ 2 (1 – x)
  2. Solve the inequality and represent it on the number line: 3 (1 – x) < 2 (x + 4).
  3. Calculate the pair of consecutive even positive integers, both of which are larger than 5, such that their total is less than 23.
  4. Solve the inequality graphically in the two-dimension plane: 3y – 5x < 30.
  5. Solve the system of linear inequalities: x + y ≤ 6, x + y ≥ 4.

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