Graphing Inequalities on Number Lines

Graphing inequalities on number lines can be extremely useful in solving mathematics, science, statistics, etc. Graphing inequalities give us a better idea about the variation in a particular parameter in a real-time scenario.  

We know equations in one variable and two variables. We have also solved several word problems by changing them into mathematical equations. However, we have only dealt with the ‘=’ sign in the equations until now. However, there are some statements in which signs like less than (<), more significant than (>), less than or equal to and greater than or equal to are also involved. These signs are called ‘inequalities’. 

Rules for Solving Inequalities

 Equal numbers can be added or subtracted from both sides of the equation.

1. Both sides of the equations can be multiplied or divided by the same number. Zero cannot be used to multiply or divide.

The rules mentioned above are the basics for solving inequality. 

 Procedure for Solving the Inequality in one Variable and their Graphical Representation

Graphing inequalities on number lines is carried out for both one and two-variable equations. In this section, we explain how to plot a graph for inequality in one variable with the help of an example:

Suppose we have this question: 

Solve 7x+3<5x+9 and graphically represent the solution on the number line.

Solution: 

7x+3<5x+9 

Simplifying the above equation,

2x<6 or x<3

Now, we have the final answer as x<3. Thus, we can plot this solution on the number line in the following way.

• Draw a number line using a scale and pencil. Put arrows at the end of the line. These arrows show that the line is infinite.

• Now draw a horizontal line on the infinite line and mark it as 0.

• Remember, the numbers to the left of zero are negative. While the numbers to the right of zero are positive. 

• Put the numbers on the number line at an equal distance from each other. 

• Now draw a hollow circle on 3. Once you have marked 3 on the number line, shade the area less than 3 on the number line. 

• The graphical representation of the equation is now complete.

Rules to follow while representing a one-variable equation on the number line.

• The line used for representing the solution should be infinite. An infinite line is shown by putting arrows at the end.

• The numbers to the left of zero are negative and positive to the right.

• Use a hollow circle for representing the solution for x> a or x<a.

• Use a dark circle for representing the solution for x≤a or x≥a.

• Shade the solution area to make it noticeable. 

Graphical solution for Linear Inequalities in two variables.

In the previous section, we learned graphing inequalities on the number line for a one-variable equation. In this section, we will learn about solving a two-variable equation graphically.

Let us consider an example: 

• Solve: x+y≥5 and x-y≤3 are the two inequalities. Solve these inequalities graphically.

• To solve this equation, we will solve the two inequalities simultaneously.

Solution of the first equation:

• To solve the first equation graphically, we will begin by drawing a graph x + y = 5.

• We will use the hit and trial method to get the graph’s points. Remember, it is best to use simple digits such as 0 and 1.

x+y=5

Assuming x = 0

0 + y = 5

Therefore, y =5

• So the first point of the line x + y = 5 is (0, 5). Similarly, we will find the second line because two points must construct a line.

Similarly,

Assuming y = 0 

0 + x = 5 

Therefore x = 5.

• The second point of the line x + y = 5 is (5, 0).

Solution of the second equation 

• Similarly, we will get the solutions for x – y = 3

Assuming x = 0, then,

-y = 3, therefore y= -3 and the first point of the equation is (0, -3)

Assuming y = 0, then

Then x = 3, hence the second point of the equation is (3, 0).

• So the point of two lines are as follows:-

x+y=5;0, 5and 5, 0

x-y=3;0, -3and 3, 0

Now we will draw the two lines on the graph paper.

Steps for Drawing Two Lines on the Graph Paper

• Draw x and y axes on graph paper and put the arrow at the end of the axes.

• Now divide the axes into equal sections using an appropriate scale, such as 10 divisions = 1 unit on both axes.

• Now plot the points (0, 5) and (5, 0) on the graph and connect them with the help of a line. Moreover, put the equation on the top of the line. 

• Similarly, plot the points (0, -3) and (3, 0) on the graph and connect them with the help of a line. Write the equation on the top of the line.

• The line divides the graph into two sections. Now choose any point from any of the sections. For example, we are choosing (0, 0).

• Put (0, 0) in the first equation. When putting (0,0) in the first equation, you will observe that. 

x+y≥5

0≥5

It is not true; hence, we will shade the area that does not contain (0, 0) above the x + y = 5.

• Similarly, we will find that when we put (0, 0) in the second equation. 

0 ≤3. 

• It is true; hence, we will shade the area consisting of the point (0, 0).

• The solution of the two systems of equations will be the common shaded area.