Linear Inequalities

Linear Inequalities in Algebra are defined as the mathematical statements that are formed by combining linear algebraic expressions with inequalities. A linear algebraic expression in an expression with degree one. Linear inequalities can be easily represented using various methods that are discussed in the article below. We have to study Linear Inequalities in Class 11.

In this article, we will learn about, Linear Inequalities, Solving Linear Inequalities, Examples of Linear Inequalities, and others in detail.

What are Linear Inequalities?

Linear Inequality expresses the comparison between two or more quantities in a mathematical form or as an expression. It must be noted that in a linear inequality, at least one quantity that is being compared is a polynomial.

We make use of various symbols to represent the linear inequality in mathematics. These symbols and their meaning is summarized in the following table:

Symbol	Meaning	Example
>	Greater Than	x > 3
<	Less Than	x < 9
≠	Not equals to	x ≠ 10
≥	Greater than or equal to	2x + 5 ≥ 11
≤	Less than or equal to	x ≤ 2x – 12

Here > and < are strict inequality symbols while ≥ and ≤ are not strict inequality symbols.

Linear Inequalty Definition

Genrally linear inequalities are considered as the inequalities that are formed using the linear expressions. A linear inequality can have one, two or more variables. For example, a linear inequality in one variable is x < 6, and a linear inequality in two variable is, x + z > 11.

Linear Inequalities Examples

Following are some examples of linear inequalities with their meaning:

Linear Inequality	Meaning
x > 5	x is greater than 5
x < 6	x is less than 6
x ≥ 1	x is greater than or equal to 1
x ≤ 0	x is less than or equal to 0
x ≠ 1	x is not equal to 1

Linear Inequalities Rules

All the mathematical operations i.e. addition, subtraction, multiplication and division are applicable to linear inequalities also. Let us see how to perform these operations on linear inequalities below:

Addition Rule

This rule states that if equal value is added to both sides of the linear inequality, then the meaning of new inequality is also the same as before and the comparison symbol does not change.

For example consider a linear inequality x < 5. If we add 5 to both sides of it, we get

x + 5 < 5 + 5

x + 5 < 10

Thus we see that adding an equal value to both sides of linear inequality does not change the comparison symbol.

Subtraction Rule

This rule states that if equal value is subtracted from both sides of the linear inequality, then the meaning of new inequality is also the same as before and the comparison symbol does not change.

For example consider a linear inequality x < 5. If we subtract 5 from both sides of it, we get

x – 5 < 5 – 5

x – 5 < 0

Thus we see that subtracting an equal value to both sides of linear inequality does not change the comparison symbol.

Note: If the both sides of inequality are subtracted from a number, then the comparison symbol will change. Assume that in the above example both sides are subtracted from 5, then we will get, 5 – x < 0 which changes the meaning of inequality and is thus not correct. In order to make it correct we need to change the comparison symbol and write it as 5 – x > 0.

Multiplication Rule

This rule states that if equal value is multiplied to both sides of the linear inequality, then the meaning of new inequality is also the same as before and the comparison symbol does not change.

For example consider a linear inequality x < 5. If we multiply 5 to both sides of it, we get

5x < 5×5

5x < 25

Thus we see that multiplying an equal value to both sides of linear inequality does not change the comparison symbol.

Note: If the both sides of inequality are multiplied by a negative number, then the comparison symbol will change. Assume that in the above example both sides are multiplied by -5, then we will get, -5x < -25 which changes the meaning of inequality and is thus not correct. In order to make it correct we need to change the comparison symbol and write it as -5x > -25.

Division Rule

This rule states that if both sides of the linear inequality are divided by equal value, then the meaning of new inequality is also the same as before and the comparison symbol does not change.

For example consider a linear inequality x < 5. If we divide by 5 sides by 5, we get

x/5 < 5/5

x/5 < 1

Thus we see that dividing both sides by an equal value does not change the comparison symbol.

Note: If an equal value is divided by both sides of inequality, then the comparison symbol will change. Assume that in the above example 5 is divided by both sides, then we will get, 5/x < 1 which changes the meaning of inequality and is thus not correct. In order to make it correct we need to change the comparison symbol and write it as 5/x > 1.

Also if the both sides of inequality are divided by a negative number, then the comparison symbol will change.

How to Solve Linear Inequalities?

There are generally two types of linear inequalities that are,

• Linear Inequalities in One Variable
• Linear Inequalities in Two Variables

There are various methods to solve these two types of linear inequalities and that includes, solving algebraicaly, graphical solution of linear equation and others, etc.

Linear Inequalities in One Variable

The linear inequalities which deal with only one variable are called Linear Inequalities With One Variable. For example x >5.

In order to solve the linear inequality with variables on one side following steps are followed:

• Use the rules of inequality mentioned above to isolate the variable on one side.
• The inequality so obtained is the required answer and tells the value of variable.

For example: Consider the inequality x+10 < 7. This can be solved as:

• Subtract 10 from both sides to get x+10-10<7-10
• Thus, we get x<-3.

Hence, x < -3 is the required value of x. This is a strict inequality.

Examples of Linear Equation in One Variable

Various examples of linear equation with one variable are,

• x > 11
• y < -4
• z < 8, etc

Linear Inequalities in Two Variables

The linear inequalities which deal with two variables are called Linear Inequalities with Two Variables. For example x – y > 5 and x + y > 4. This is also called as system of linear inequalities.

In order to solve the linear inequality with two variables, it is necessary to have at least two linear inequalities with the same variables. These type of linear inequalities can be solved only through graphing.

Examples of Linear Equation with Two Variable

Various examples of linear equation with two variable are,

• x + y > 11
• y – z < -4
• z + x < 8, etc

Graphing Linear Inequalities

Graphing linear inequalities involves representing the solutions to the inequality on a coordinate plane. Both inequalities, whether involving one variable or two, can be plotted on the two-dimensional coordinate plane with the help of various algebraic methods. We will discuss here graphs for:

• Linear Inequalities with One Variable
• Linear Inequalities with Two Variable

Let’s discuss graphing these linear inequalities in detail.

Graphing Linear Inequalities with One Variable

Linear inequalities in one variable are represented on a number line. The basic steps followed to represent a linear inequality with one variable on a number line are:

• Solve the linear equality in one variable using above method.
• If the linear inequality is a strict inequality then use open interval to represent the set of numbers that satisfy the linear inequality. An open interval is represented using ( ) parentheses.
• If the linear inequality is not a strict inequality then use closed interval to represent the set of numbers that satisfy the linear inequality. A closed interval is represented using [ ] parentheses.

In the above example, the linear inequality after solving can be represented in the following open interval (-∞,-3) as it is a strict linear inequality. This can be plotted on a number line as follows:

In this number line an open circle on the value obtained after solving the linear inequality is used to denote strict inequality. The direction of the green arrow shows the direction in which the numbers on the number line will satisfy the given inequality.

Graph of Linear Inequalities in Two Variables

The graph of a system of linear inequalities is plotted using cartesian coordinate system which has X -axis and Y-axis. Following steps are followed to solve them through graphs:

• Replace all the inequality symbols with = sign so as to obtain an equation of line.
• Plot the lines on the graph.
• Select a point on the LHS or RHS side of the line on the graph. If it satisfies the linear inequality, then mark the region on that side where the point lies. Else, mark the region on the other side of the line.
• Repeat this step for all the linear inequalities given to us.
• Once the regions have been marked, shade the region that is common to all the linear inequalities.
• The common shaded region is the solution to the given system of linear inequalities. In case there is no common area, then there is no solution the system of linear inequalities.

Let us understand this with an example:

Consider the following system of linear inequalities x – 2y < -1 and 2x – y > 1

Solution:

Step 1: Replace all the inequality symbols with = sign so as to obtain an equation of line.

x – 2y = -1 and 2x – y = 1

Step 2: Plot the lines on the graph as follows:

Step 3:

• Select the point (2, 2) for line x – 2y = 1. Check if this point satisfies linear inequality or not. As 2 – 2(2) = -2 < -1, the point satisfies the linear inequality. Thus the area on the side of the point is marked.
• Select the point (2, 1) for line 2x – y = 1. Check if this point satisfies linear inequality or not. As 2(2) – 1 = 3 > 1, the point satisfies the linear inequality. Thus the area on the side of the point is marked.

Step 4: The area common to both the lines is shaded in purple as seen in the diagram.

Thus all the points that lie in the shaded region satisfy the linear inequality.

System Of Linear Inequalities

When we have multiple linear inequalities with same variables, then they form a system of linear inequalities. The system of linear inequalities is solved through the graph method as discussed above.

In order to solve the system of linear inequalities, it is necessary to have at least two linear inequalities if there are 2 variables or in other words, the number of linear equalities must be equal to the number of variables. Let us understand how to solve the system of linear inequalities with an example.

Example: Consider the following system of linear inequalities, x – 2y > -1 and 2x – y < 1

Solution:

Step 1: Replace all the inequality symbols with = sign so as to obtain an equation of line.

x – 2y = -1 and 2x – y = 1

Step 2: Plot the lines on the graph as follows:

Step 3:

Select the point (2, 2) for line x – 2y = 1. Check if this point satisfies linear inequality or not. As 2 – 2(2) = -2 > -1, the point satisfies the linear inequality. Thus the area on the side of the point is marked.

Select the point (2, 1) for line 2x – y = 1. Check if this point satisfies linear inequality or not. As 2(2) – 1 = 3 < 1, the point satisfies the linear inequality. Thus the area on the side of the point is marked.

Step 4: The area common to both the lines is shaded in purple as seen in the diagram.

Thus all the points that lie in the shaded region satisfy the linear inequality.

Applications of Linear Inequalities

Linear inequalities has various applications such as:

• They are used to model real life business problems.
• They are used in a game called inequality sudoku.
• They also find applications in astronomy and space research.
• They are also used in business to make decisions such as maximising the profit and minimising the cost of production.

Read More,

• Compound Inequalities
• Algebraic Identites
• Trigonometric Identites

Examples on Solving Linear Inequalities

Example 1: Solve the inequality 2x + 3 < 5.

Solution:

Given 2x + 3 < 5

• Subtract 3 from both sides

2x < 2

• Divide both sides by 2

x < 1

Thus x < 1 is the required inequality.

Example 2: Solve the inequality x + 9 < 5x.

Solution:

Given x + 9 < 5x

• Subtract 9 from both sides

x < 5x – 9

• Subtract x from both sides

x – x < 5x – 9 -x

0 < 4x – 9

• Add 9 to both sides

9 < 4x

• Divide both sides by 4

9/4 < x

Thus x > 9/4 is the required inequality.

Example 3: Solve the inequality x + 3 < 5 + 2x.

Solution:

Given x + 3 < 5 + 2x

• Subtract x from both sides

x + 3 – x < 5 + 2x – x

3 < 5 + x

• Subtract 5 from both sides

-2 < x

Thus x > -2 is the required inequality.

Example 4: Solve the inequality 3 < x – 8.

Solution:

Given 3 < x – 8

• Add 8 to both sides

11 < x

Thus x > 11 is the required inequality.

Example 5: Solve the inequality x/5 + 3 < 8.

Solution:

Given x/5 + 3 < 8

• Subtract 3 from both sides

x/5 + 3 – 3 < 8 – 3

x/5 < 5

• Multiply both sides by 5

x < 25

Thus x < 25 is the required inequality.

Example 6: Solve the inequality 34 < -x + 7.

Solution:

Given 34 < -x + 7

• Subtract 7 from both sides

27 < -x

• Multiply both sides by -1. Note that the sign of inequality will reverse now.

-27 > x

Thus x < -27 is the required inequality.

Example 7: Graph the inequality x ≥ 6.

Solution:

Given x ≥ 6

The numbers that satisfy this linear inequality are represented in the closed interval [6, ∞) as it is not a strict inequality. The graph for this inequality is shown below:

Example 8: Solve the system of linear inequalities y ≤ x – 1 and y < –2x + 1.

Solution:

Given y ≤ x – 1 and y < –2x + 1

• Step 1: Replace all the inequality symbols with = sign so as to obtain an equation of line.

y = x – 1 and y = -2x + 1

• Step 2: Plot the lines on the graph as follows:

• Step 3:
  • Select the point (3, 1) for line y = x – 1. Check if this point satisfies linear inequality or not. As 1 ≤ 3 – 1 = 2, the point satisfies the linear inequality. Thus the area on the side of the point is marked.
  • Select the point (-1, 1) for line y = -2x + 1. Check if this point satisfies linear inequality or not. As 1 < -2(-1)+1 = 3 , the point satisfies the linear inequality. Thus the area on the side of the point is marked.
• Step 4: The area common to both the lines is shaded in purple as seen in the diagram.
• Thus all the points that lie in the shaded region satisfy the linear inequality.

Solve Linear Inequalities

Q1. Solve x + y ≤ 11 and x ≥ y – 1

Q2. Solve 3y ≤ x – 12 and x < 3x + 5

Q3. Solve 5y ≥ x – 18 and y < x + 1

Q4. Solve y >x – 1 and y ≥ –2x + 1

Linear Inequalities – FAQs

1. What is Linear Inequality Class 11?

Linear inequality expresses the comparison between two or more quantities in a mathematical form or as an expression.

2. What is meaning of Symbols > and ≤ in Linear Inequality?

Symbols > and ≤ refer to greater than and less than equal to in linear inequalities.

3. What is the Formula for Linear Inequalities?

The general form to represent the linear inequalities in two variables is,

• ax + by > c
• ax + by < c, etc.

4. What is the Purpose of Linear Inequalities?

System of Linear Inequalties are used to determine the maximum and minimum value of the various situations given as algebric expressions and explained with some constrains.

5. What are Inequality Symbols?

The five common symbols used to represent the inequalities are,

• Greater Than Symbol(>)
• Less Than Symbol(<)
• Greater Than Equal to Symbol(≥)
• Less Than Equal to Symbol(≤)
• Does not Equal to Symbol(≠)

6. What are Examples of Linear Inequality?

Examples of linear inequality are,

• x + 1 > 5
• x – 1 < 3
• x + y ≤ 12
• x – y ≥ 5, etc.

7. What Are the Real-Life Uses of Linear Inequalities?

Real-Life Uses of Linear Inequalities are,

• They are used to model real life business problems.
• They are used to solve various mathematical puzzels, etc.

8. How can we Solve Linear Inequalities in Two Variables?

Linear inequalities in two variables can be solved with the help of graphs and determining the common region that satisfies the given inequalities.

9. How are Quadratic Inequalities Different From Linear Inequalities?

Inequalities with algebraic expressions of degree 2 are called quadratic inequalities whereas inequalities with algebraic expressions of degree 1 are called linear inequalities.

10. How to Draw Linear Inequalities Graph?

We can draw a Linear Inequalities graph by first solving the given inequalities and finding the points. These points are plotted and then graph is drawn. Detailed steps has been mentioned in the article.