Linear Inequalities with Definitions, Formulas, Signs, Rules, Graphical Representation and Solved Examples
Saurav Sinha
| Updated: Apr 22, 2021 19:06 IST
What are Linear Inequalities?
Linear inequalities imply when 2 real numbers or algebraic expressions are associated by the symbol ‘<’, ‘>’, ‘≤’ and ‘≥’ which forms an inequality.
- An inequality statement with ‘<’, and ‘>’ signs forms strict inequalities.
- An inequality statement with ‘≤’ and ‘≥’ signs forms slack inequalities.
- An inequality statement with one variable is termed as linear inequality in one variable.
- An inequality statement with two variables is termed as linear inequality in two variables.
Linear inequalities in one variable are formed when 2 linear algebraic expressions in one variable are related by the symbol ‘<’, ‘>’, ‘≤’ and ‘≥’.
Example of a linear inequality in one variable: 3x + 5 < 10.
linear inequalities in two variables are formed when 2 linear algebraic expressions in two variables are related by the symbol ‘<’, ‘>’, ‘≤’ and ‘≥’.
Example of a linear inequality in two variables: 3x + 5y ≥ 20.
Signs of Linear Inequalities
The signs of linear inequalities are as follows:
| Inequality | Symbol |
| less than | < |
| greater than | > |
| less than or equal | ≤ |
| greater than or equal | ≥ |
| not equal to | ≠ |
What are different Linear inequalities rules?
Let us study the linear inequalities rules one by one below:
Addition Rule of Linear Inequalities
Addition rule of linear inequalities states that when equal numbers are added on both sides of inequalities then the sign of inequality does not change. Consider the below linear inequalities example to understand the concept.
Addition rule of linear inequalities example:
- Let 3x + 5 < 10 be the given inequality.
- Now by adding 5 from both sides of the inequality, we get:
- 3x + 5 + 5 < 10 + 5
- ⇒ 3x + 10 < 15
Subtraction Rule of Linear Inequalities
Subtraction rule of linear inequalities states that when equal numbers are subtracted on both sides of inequalities then the sign of inequality does not change. Check the below linear inequalities example to understand the concept.
Subtraction rule of linear inequalities example:
- Let 3x + 5 < 10 be the given inequality.
- Now by subtracting 5 from both sides of the inequality, we get:
- 3x + 5 – 5 < 10 – 5
- ⇒ 3x < 5
Multiplication Rule of Linear Inequalities
Multiplication rule of linear inequalities states that when both sides of the inequality are multiplied by the same positive number then the sign of inequality does not change. Whereas, if both sides are multiplied by the same negative number then the sign of inequality gets reversed.
Multiplication rule of linear inequalities example:
- Let 3x < 5 be the given inequality.
- Now, when both sides of the inequality are divided by 3, we get x < 5 / 3.
- Now, when both sides of the inequality are multiplied by 3, we get 3x < 15 / 9.
Division Rule of Linear Inequalities
Division rule of linear inequalities states that when both sides of the inequality are divided by the same positive number then the sign of inequality does not change. Whereas, if both sides are divided by the same negative number then the sign of inequality gets reversed.
Example:
- Let 3x < 5 be the given inequality.
- Now, when both sides of the inequality are divided by 3, we get x < 5 / 3.
- Whereas if both sides of the given inequality are divided by – 3, we get – x > – 5 / 3.
Learn more about Linear Inequalities symbols.
Linear Inequalities In One Variable
Linear inequality in one variable forms when 2 linear algebraic expressions in one variable are related by the symbol ‘<’, ‘>’, ‘≤’ and ‘≥’.
Example 9x + 5 < 15.
Linear Inequalities In Two Variables
Linear inequality in two variables forms when 2 linear algebraic expressions in two variables are related by the symbol ‘<’, ‘>’, ‘≤’ and ‘≥’.
Linear Inequalities In Two Variables Solved Example: 2x + y > 6
Given: 2x + y > 6
Step 1: Replace the inequality present in the given inequation with “=” we get 2x + y = 6.
Step 2: Represent the equation 2x + y = 6 on a 2D plane as shown below:
Step 3: Now, put x = 0 and y = 0 in the given inequality: 2x + y > 6 we get 0 + 0 > 6, and this is false.
Hence, shade the portion which does not contain the point (0, 0) as shown in the figure above.
The shaded portion in the above-given diagram represents the solution to the given inequality.
Now, you have learned Linear Inequalities so you can attempt Linear Inequalities MCQs.
Graphical Representation of Linear Inequalities Question with Solution
Let’s understand the graphical representation of linear inequalities in one variable with the help of an example.
Solved Example: 5x – 3 < 3x + 1, where x ∈ R
Given: 5x – 3 < 3x + 1
- Step 1: Now by subtracting 3x from both sides of the given inequality, we get;
- Step 2 :⇒ (5x – 3) – 3x < (3x + 1) – 3x
- Step 3 : ⇒ 2x – 3 < 1
- Step 4: Now by adding 3 on both sides of the above inequality we get; ⇒ 2x < 4
- Step 5: Now by dividing both sides of the above inequality by 2, we get ⇒ x < 2
- Step 6 : ∵ x ∈ R ⇒ x = (- ∞, 2)
The graphical representation of the solution of the inequality 5x – 3 < 3x + 1, where x ∈ R i
Therefore, any solution of an inequality in one variable is a value of the variable which makes it a true declaration. This indicates we have found the solutions to the inequality by trial and error method.
Learn about Linear Inequalities in this video
| If you are checking Linear Inequality article, also check the related maths articles in the table below: | |
| Less than or Equal to Symbol | Linear Equation In Two Variables |
| Linear Arrangement Reasoning | Speed, Time and Distance |
| Polynomial Identities Algebra | Linear Equations in Two Variables |
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Linear Inequalities FAQs
Q.1 What are various signs used to represent inequality?
Ans.1 The signs used are as follows; ‘<’ (less than), ‘>’ (greater than), ‘≤’ (less than or equal), ‘≥’ (greater than or equal), and ‘≠'(not equal to).
Q.2 What is one variable linear inequality?
Ans.2 An inequality statement with one variable is termed as linear inequalities in one variable.
For example: 7x + 8< 10.
For example: 7x + 8< 10.
Q.3 What is two-variable linear inequality?
Ans.3 An inequality statement with two variables is termed as linear inequalities in two variables.
For example: 9x + 4y ≥ 50.
For example: 9x + 4y ≥ 50.
Q.4 What are strict inequalities?
Ans.4 Inequality statement with ‘<’, and ‘>’ signs forms strict inequalities.
Q.5 What are slack inequalities?
Ans.5 Inequality statement with ‘≤’ and ‘≥’ signs forms slack inequalities.
Q.6 What is linear inequality definition?
Ans.6 Linear inequalities are the expressions where any two values are compared by the inequality symbols such as, ‘<‘, ‘>’, ‘≤’ or ‘≥’.
