Linear Inequalities

A linear inequality is similar to a linear equation, but instead of an equal sign (=), it uses an inequality sign (such as <, ≤, >, ≥ ). These inequalities describe a region of the graph, rather than just a line.

Forms of Linear Inequalities:

Linear inequalities are formed by combining linear algebraic expressions with inequalities. A linear algebraic expression has a degree of one, meaning it involves terms of degree one.

• Linear inequality in one variable: For example, x < 6.
• Linear inequality in two variables: For example, x + z > 11.

In linear inequalities, at least one quantity being compared must be a polynomial.

Inequality Symbols and their Meanings:

• Strict inequality symbols: > and < (do not include the boundary points).
• Non-strict inequality symbols: ≥ and ≤ (include the boundary points).

Linear Inequalities Examples

Following are some examples of linear inequalities with their meaning:

Linear Inequalities Rules

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

Adding or Subtracting the Same Value

When you add or subtract the same number (or expression) to both sides of the inequality, the inequality sign remains the same.

Example:

x + 3 > 7 ⇒ x > 4
2x − 5 ≤ 9 ⇒ 2x ≤ 14 ⇒ x ≤ 7

Multiplying or Dividing by a Positive Number

When you multiply or divide both sides of the inequality by a positive number, the inequality sign remains the same.

Example:

3x < 12 ⇒ x < 4
\frac{2x}{3} \geq 4 \Rightarrow x\geq 12

Multiplying or Dividing by a Negative Number

When you multiply or divide both sides of the inequality by a negative number, reverse the inequality sign.

Example:

−2x > 6 ⇒ x < −3 (reverse the inequality sign)
\frac{-3x}{4} \leq 9 \quad \Rightarrow \quad x \geq -12 \quad (reverse the inequality sign)

Combining Inequalities

When you have an inequality with a common variable on both sides, you can combine them by solving the inequality.

Example: 2x − 3 ≥ 5 and x + 1 < 6

Solve each inequality:

• 2x ≥ 8 ⇒ x ≥ 4
• x < 5

Combine: 4 ≤ x < 5

Inequalities Involving Absolute Values

For inequalities with absolute values, treat them as two separate inequalities based on the definition of absolute value.

Example: ∣x − 3∣ ≤ 5|

This splits into two inequalities: −5 ≤ x − 3 ≤ 5

Solve: −2 ≤ x ≤ 8

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 algebraically, graphical solutions, 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, maximizing 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.

Check, Graphical Solution of Linear Inequalities in Two Variables 

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.

Check, Systems of Inequalities

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 maximizing the profit and minimizing the cost of production.

Read More,

• Inequalities: Definition, Properties, Rules, Types
• Graphical Solution of Linear Inequalities in Two Variables
• Trigonometric Identities
• Algebraic Identities
• Compound Inequalities

Linear Inequalities Solved Examples

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 + 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 3: 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 4: 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 5: 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.

Check, Practice Questions on Linear Inequalities