Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
Welcome to our Math lesson on Solving Linear Inequalities in Two Variables, this is the sixth lesson of our suite of math lessons covering the topic of Solving Linear Inequalities, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
When dealing with linear equations with two variables, we explained that a single linear equation gives an infinity of number pairs that are all solutions for the given equation. We cannot expect anything else when dealing with linear inequalities as they usually contain more possible solutions than the corresponding equations. In other words, the number of possible solutions of the linear inequality
is much greater than that of the corresponding equation
despite both having an infinite number of possible solutions.
Linear inequalities with two variables are better understood when using the graph method, which we will explain in tutorial 10.3. However, we can use analytical methods to solve linear inequalities as well. Just one thing to remember: we have to choose a range of allowed values for the dependent variable x and based on this range calculate the corresponding range of values for the dependent variable y. For example, if we have to find the solution set of the linear inequality
if the variable x takes the values from the segment [2, 7], we must solve two linear inequalities with one variable like those discussed in the previous paragraphs, where the variable x is replaced by the two limit values of the given segment. In this way, the two linear inequalities are solved only for y.
Thus, for x = 2, we obtain
and for x = 7, we obtain
Thus, since the values of y must be limited between the values found for the two limit values of x, we obtain for the solution set of the inequality the values of y that extend from 5 and up for the minimum value of x, to 20 and up for the maximum value of x, without including these limit values.
We can check whether a certain number pair belongs to the solution set of a linear inequality or not by substituting the values in the inequality. In this way, after doing all operations, we see whether the final version of the simplified inequality is true or not. Let's take an example to clarify this point.Check whether the number pairs (3, 1), (-3, 4) and (0, 2) belong to the solution set of the inequality
First, let's write the inequality in such a way as to isolate y on the left side and express all the rest of the terms on the right side of the inequality symbol. Thus,
Looking at this inequality from right to left yields
Applying the properties of inequalities yields
Now, let's check whether the given number pairs are solutions for the above inequality or not. Thus, for the number pair (3, 1) (x = 3 and y = 1) we have
Hence, the number pair (3, 1) is not a solution for the given linear inequality.
As for the number pair (-3, 4) (x = -3 and y = 4), we have
Hence, the number pair (-3, 4) is a solution for the given linear inequality.
Last, for the number pair (0, 2) (x = 0 and y = 2), we obtain
Hence, the number pair (0, -4) is not a solution for the given linear inequality.
Enjoy the "Solving Linear Inequalities in Two Variables" math lesson? People who liked the "Solving Linear Inequalities lesson found the following resources useful:
Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
We hope you found this Math tutorial "Solving Linear Inequalities" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.