Graphical Solution of Linear Inequalities in Two Variables

Linear inequalities in two variables denote the inequality between any two expressions of algebraic nature. These inequalities include two unique variables. This article will explain how to find the graphical solution of linear inequalities in two variables. 

Linear inequalities in two variables help indicate the unequal relationship between two expressions of algebraic nature. These two expressions consist of two distinct or readily distinguishable variables. The symbols used to represent a linear inequality with two variables include less than, greater than, less than or equal to, and greater than or equal to. 

Let us now go deep into the study of linear inequalities in two variables.

Understanding Linear Inequalities in Two Variables

Before reading the linear inequalities in two variables PDF, it is essential to know what exactly it implies. This factor will also help you find the graphical solution of linear inequalities in two variables.

To form a linear inequality, an association of two algebraic expressions or real numbers must take place with certain symbols. These symbols can be ‘<,’ ‘>,’ ‘≤,’ or ‘≥.’ The equal to (=) symbol is not used to denote linear inequalities. Equations with strict inequalities are represented by the symbols ‘<’ or ‘>,’ while slack inequalities are characterized by the symbols ‘≤’ or ‘≥.’ 

Examples of numerical inequalities: 

• 7 < 10
• 20 > 19 

Examples of linear inequalities:

• x < y + 1
• x – y > 30 

The inequalities in two variables have various types of examples, which are follows:

• X + 2y < 5 (less than)
• X + 2y > 5 (greater than)
• X + 2y ≤ 5 (less than or equal to)
• X + 2y ≥ 5 (greater than or equal to)

Solving Problems Involving Linear Inequalities in Two Variables

One can solve problems by referring to the linear inequalities in two variables PDF. Let us now look at two examples of solving problems involving linear inequalities in two variables.

Example 1:

If Ax + By < C is a linear inequality with the two variables x and y, then an ordered pair (x, y) satisfying the inequality will be the solution.

Let us understand this concept better. 

If we have an equation 3x + 4y > 5, we can check for the values of x and y that satisfy this inequality. 

So, we must consider x = 1 and y = 2. 

Substituting the above values in the LHS of the inequality, we get:

3(1) + 4(2) = 3 + 8 =11

Since 11 > 5, the ordered pair (1, 2) satisfies the inequality 3x + 4y > 5. Hence, we can say that (1, 2) is the solution to inequality. We can further substitute other x and y to check for all possible solutions that satisfy the inequality.

Example 2:

We have to find the solution by making the following two assumptions:

• x=0 
• y=0

Now we can find the solution for the problem- x + 9y < 27.

To find the solution for this, we need a pair of values of x and y for the solution set.

Consider, x = 0

So, the problem becomes 0 + 9y < 27

Now, 9y < 27

Also, y < 3

Consider, y = 0

So, the problem becomes x + 9(0) < 27

Also, x < 27

So, (0, 3) and (27, 0) are the given inequality’s solutions.

If x = 0, then y could be lower than the value of 3, which is 0, 1, 2. The inclusion of 3 will not take place here, as it is not equal to three. 

Finally, we get the following solution set:

(0, 0), (0, 1), (0, 2).

Graphical Solution of Linear Inequalities in Two Variables

For any linear equation with two variables, there are an infinite number of solutions. The same is with linear inequalities, as there are an endless number of solutions that meet the conditions of the inequality. 

We will now look at the graphical solution of linear inequalities in two variables. 

The solutions to linear inequalities in two variables are known as ordered pairs. One can graph these ordered pairs or solution sets in the appropriate half of a rectangular coordinate plane. 

Before creating the graphical solution of linear inequalities in two variables, you must first identify the type of inequality. The kind of inequality could be less than, greater than, less than, equal to, or greater than or equal to. 

The next step is to create a graph of the boundary line. The boundary line could either be a dashed line in the case of a strict inequality or a solid line, as in the case of slack inequalities or non-strict inequalities. 

Third, you need to select a test point. Typically, (0,0) is the test point in most cases or any other point not on the boundary line. 

Next, shade the appropriate region if the test point consists of the inequality. Shade the area that contains it. If not, shade the part on the opposite side of the boundary line. 

You can also verify and shade the appropriate region with more significant test points in and out of the boundary line. The solution sets or the ordered pairs correspond to half-planes in the case of linear inequalities, while the solution sets to linear equations correspond to lines.

Conclusion

Linear inequalities in two variables help us see the inequality that exists between two expressions of algebraic nature. There must be an association of two real numbers or algebraic expressions with certain symbols such as ‘<,’ ‘>,’ ‘≤,’ or ‘≥’ to form a linear inequality. We have gone through what linear inequalities are, how inequalities exist in the case of two variables, and the graphical solution of linear inequalities in two variables. Now, you can go ahead and solve two variable linear inequalities yourself and plot the solutions on a graph.