Solutions of Linear Inequalities in Two Variables: Equations or equalities are mathematical representations of any two things that are equal. While equalities represent a relationship between two variables, inequalities represent situations that are better described by a range of values instead of a single solution. This means that, mathematically, inequalities have more than one correct answer. Similar to equalities, the graphs of inequalities may broadly be of two types: Linear inequality and nonlinear inequality.

Here, let us focus on linear inequalities and understand how to solve them.

What are Linear Inequalities?

An equality or equation is two mathematical expressions joined a \(” = ” \) symbol. This means that the LHS is equal to the RHS. When two polynomials are not equal and joined by a comparison, they are called an inequality. The symbols of inequality are listed in the table.

Symbol of Inequality	Meaning
\( \ne \)	Not equal to
\( < \)	Lesser than
\( > \)	Greater
\( \le \)	Less than or equal to
\( \ge \)	Greater than or equal to

In mathematics, linear inequality is defined as a linear function in which two values are compared using the inequality symbols. An inequality is said to be linear if the degree of the polynomial is one.

Example of linear inequalities: For a linear polynomial \(ax + b,\) the inequalities may be \(ax + b \le 0,\,ax + b \ge 0,\) or \(ax + b \ne 0,\) where \(a\) and \(b\) are real numbers.

The common method to solve inequalities is by graphing them.

Linear Inequalities in Two Variables

Linear inequalities in two variables are represented as inequality of two algebraic expressions that involve only two variables. Example: \(ax + by \le c.\)
Here, \(x\) and \(y\) are the variables, while \(a\) and \(b\) are coefficients and \(c\) is the constant. If \(ax + by\) and \(c\) are equal, then they form linear equality in two variables.

Graphing Linear Inequalities

In earlier topics, we have learnt that a linear equation in two variables has infinitely many ordered pairs as solutions. We also know that the graph of a linear equation, of the form \(ax + by = c,\) is always a straight line. This can be verified by graphing the equation on a coordinate plane.
For example, the graph of linear equality \(x + y = 2\) is given below.

On the other hand, for a linear inequality in two variables, the solution set is a region that defines a part of the coordinate plane.

Now take a look at the following graphs of inequalities.

\(x + y < 2\)	\(x + y > 2\)
\(x + y \le 2\)	\(x + y \ge 2\)

For a linear inequality, the solution set is a shaded half-plane. For \(y\) greater than inequality, the region above the line is shaded, and for \(y\) lesser than inequality, the region below the line is shaded.

Observe that the line, either dotted or solid. This line serves as a boundary on one side, while the region extends infinitely on other sides. This shows that any point that lies in the shaded region will satisfy the inequality.

Let us verify this using a sample point for each of these inequalities.

Linear Inequality	Test Point from the Shaded Region	Validity
\(x + y < 2\)	\(\left( { – 2,\, – 2} \right)\)	\( – 2 + \left( { – 2} \right) < 2\) \( – 4 < 2\) Lies in the region
\(x + y > 2\)	\(\left( {3,\,3} \right)\)	\(3 + 3 > 2\) \(6 > 2\) Lies in the region
\(x + y \le 2\)	\(\left( {0,\,0} \right)\)	\(0 + 0 \le 2\) \(0 \le 2\) Lies in the region
\(x + y \ge 2\)	\(\left( {3,\,3} \right)\)	\(3 + 3 \ge 2\) \(6 \ge 2\) Lies in the region

What about the points that lie on the line?

Take a closer look at these two graphs.

\(x + y < 2\)	\(x + y \le 2\)
Non-inclusive Boundary	Inclusive Boundary

You will see that the line is dotted in the first graph, while it is solid in the second. The dotted line represents a non-inclusive boundary, and the solid line represents an inclusive boundary.

The graph of a linear inequality is always a region, as shown in the graphs. However, the inclusion or exclusion of the line depends on the type of inequality. For a strict inequality like \( < \) or \( > ,\) the line is not included in the solution set and is represented by a dotted line. Whereas, for a slack inequality like \( \le \) or \( \ge ,\) the line is included in the solution set and is represented by a solid line.

Test Point	\(x + y < 2\)	\(x + y \le 2\)
\(\left( {4,\, – 2} \right)\)	\(4 + \left( { – 2} \right) < 2\) \(2 < 2\) Not a valid point.	\(4 + \left( { – 2} \right) < 2\) \(2 \le 2\) Valid point.

Steps to Graph a Linear Inequality

Step 1: Make the inequality into equality by changing the inequality symbol to \( = .\)
Step 2: Identify two or more points, and use the points to plot the linear graph.
a. For strict inequality such as \( \le \) and \( \ge ,\) use a solid line.
b. For slack inequality such as \( < \) and \( > ,\) use a dotted line.
Step 3: Use a test point that does not lie on the line to check if it satisfies the inequality. Use this to shade the solution set.
a. If the inequality is satisfied, shade the half-plane that has the test point.
b. If the inequality is not satisfied, shade the half-plane that does not have the test point.

System of Linear Inequalities

A system of linear inequalities in two variables includes more than one linear inequality in the same variables. A system of linear inequalities will look like this:

\({a_{11}}{x_1} + {a_{12}}{x_2} + {a_{13}}{x_3} + …. +{a_{1n}}{x_n} < {b_1}\)
\({a_{21}}{x_1} + {a_{22}}{x_2} + {a_{23}}{x_3} + …. +{a_{2n}}{x_n} < {b_2}\)
\({a_{31}}{x_1} + {a_{32}}{x_2} + {a_{33}}{x_3} + …. + {a_{3n}}{x_n} < {b_3}\)
\( \vdots ~~~~~~~~~~ \vdots ~~~~~~~~~~ \vdots ~~~~~~~~~~~~~~~~~~~~~~ \vdots ~~~~~~~~~~ \vdots \)
\({a_{m1}}{x_1} + {a_{m2}}{x_2} + {a_{m3}}{x_3} + …. + {a_{mn}}{x_n} < {b_m}\)
Here,
\({a_{11}},{a_{12}},{a_{13}},… \to \) coefficients of the linear inequality system
\({x_1},{x_2},{x_3},….. \to \) variables of the linear inequality system
\({b_1},{b_2},{b_3},…. \to \) constants of the linear inequality system
The solution set for a system of linear inequalities is the intersecting regions of the solution sets of the individual inequalities in that system.
Example of a system of linear inequalities:\(3x + y > 5\) and \(x – y \le 2.\)
The graph of this system of linear inequalities is as shown below.

Solved Problems – Solutions of Linear Inequalities in Two Variables

Q.1. Solve: \(x + 9y < 27.\)
Ans: \(x + 9y < 27\)
\(9y < 27 – x\)
\(y < 3 – \frac{x}{9}\)

When \(x = 0\)	When \(y = 0\)
\(0 + 9y < 27\) \(y < \frac{{27}}{9}\) \(y < 3\)	\(x + 9\left( 0 \right) < 27\) \(x + 0 < 27\) \(x < 27\)
\(\left({0,3} \right)\)	\(\left({27,0} \right)\)

Since this is a slack and \(y\) is lesser than inequality, the solution set will be lesser than \(3.\)
Hence, the solution set includes \(\left({0,0} \right),\left({0,1} \right),\) and \(\left({0,2} \right).\) These points will lie in the solution region.

Q.2. Graph the solution set of \(3x + y > 1.\)
Ans: \(3x + y > 1.\) can be written an \(y > 1 – 3x\)

\(x\)	\(0\)	\(1\)	\( – 1\)
\(y\)	\(y > 1 – 3\left( 0 \right)\) \(y > 1\)	\(y > 1 – 3\left( 1 \right)\) \(y > – 2\)	\(y > 1 – 3\left({ – 1} \right)\) \(y > 4\)
\(x,y\)	\(\left({0,1} \right)\)	\(\left({1, – 2}\right)\)	\(\left({ – 1,4} \right)\)

Since this is a slack inequality, we mark the graph with a dotted line.

To find the half-plane that is a solution set, use the coordinates of the test point \(\left({2,4} \right).\)
\(4 > 1 – 3\left( 2 \right)\)
\(4 > 1 – 6\)
\(4 > – 5\)
\(\therefore \) The half-plane that holds \(\left({2,4} \right)\) is the solution set of the given inequality \(y > 1 – 3x\)
The solution set of the inequality \(y > 1 – 3x\) is as shown below.

Q.3. Brook’s mother hands over \(₹7\) to him for chocolates. A milk chocolate costs \(₹2\) while a nuts chocolate costs \(₹3.\) Let x be the number of milk chocolates and y be the number of nuts chocolates. Form an inequality corresponding to the above situation and graph the inequality.
Ans: Cost of milk chocolates \( = 2x\)
Cost of nuts chocolates \( = 3y\)
Total amount in hand \(=₹ 7\)
\(\therefore \) Cost inequality \( = 2x + 3y \le 7\)
\( \Rightarrow 3y \le 7 – 2x\)
\( \Rightarrow y \le \frac{{7 – 2x}}{3}\)

\(x\)	\(0\)	\(2\)	\( – 1\)
\(y \le \frac{{7 – 2x}}{3}\)	\(y \le \frac{{7 – 2\left( 0 \right)}}{3}\) \(y \le \frac{{7 – 0}}{3}\) \(y \le \frac{7}{3}\) \(y \le 2.3\)	\(y \le \frac{{7 – 2\left( 2 \right)}}{3}\) \(y \le \frac{{7 – 4}}{3}\) \(y \le \frac{3}{3}\) \(y \le 1\)	\(y \le \frac{{7 – 2\left( { – 1} \right)}}{3}\) \(y \le \frac{{7 + 2}}{3}\) \(y \le \frac{9}{3}\) \(y \le 3\)
\(\left({x,y} \right)\)	\(\left({0,2.3} \right)\)	\(\left({2,1} \right)\)	\(\left({ – 1,3} \right)\)

Since this is a strict inequality, we mark the graph with a solid line.

To find the half-plane that is a solution set, use the coordinates of the test point \(\left({3,3} \right).\)
\(2\left( 3 \right) + 3\left( 3 \right) \le 7\)
\(6 + 9 \le 7\)
\(15 \le 7\)
\(\therefore \) The half-plane that does not hold \(\left({3,3} \right)\) is the solution set of the given inequality \(2x + 3y \le 7.\)
The solution set of the inequality \(2x + 3y \le 7\) is as shown below.

Q.4. Graph the solution set:
\(\left\{ {\begin{array}{*{20}{c}} { – 2x + y > – 4}\\ {3x – 6y \ge 6} \end{array}} \right.\)
Ans:

\( – 2x + y > – 4\)	\(3x – 6 \ge 6\)
\(y > – 4 + 2x\)	\(y \le \frac{{3x – 6}}{6}\) \(y \le \frac{x}{2} – 1\)
Since this is a slack inequality, We mark the graph with a dotted line.	Since this is a strict inequality, We mark the graph with a solid line.
For a \(y\) is greater than inequality, the region above the line is the solution set.	For a \(y\) is lesser than inequality, the region below the line is the solution set.

The graph for the system of linear inequalities will appear as shown.

The solution set for the system of linear equations is the intersecting region to all the inequalities in that system.

Q.5. Show that the given system of linear inequalities has no solution:
\(x + 2y \le 3\)
\(3x + 4y > 12\)
\(x \ge 0\)
\(y \ge 1\)

Ans:

\(x + 2y \le 3\)	\(3x + 4y > 12\)
\(y \le \frac{{3 – x}}{2}\)	\(y > \frac{{12 – 3x}}{4}\)
Since this is a strict inequality, We mark the graph with a solid line.	Since this is slack inequality, We mark the graph with a dotted line.
For a \(y\) is lesser than inequality, the region below the line is the solution set.	For a \(y\) is greater than inequality, the region above the line is the solution set.
\(x \ge 0\)	\(y \ge 1\)

\(x \ge 0\)	\(y \ge 1\)
Since this is a strict inequality, we mark the graph with a solid line.	Since this is a strict inequality, we mark the graph with a solid line.

For a \(y\) is greater than inequality, the region above the line is the solution set. These are marked as shown below:

The graph for the system of linear inequalities will appear as shown.

The solution set for the system of linear equations is the intersecting region to all the inequalities in that system.

As you can see, there are no intersecting regions for the four inequalities.

\(\therefore \) There is no solution for the given system. In other words, the solution is a null set.

Summary

In this article, we have learnt to identify linear inequalities in two variables. We have also learnt the various inequalities that are used and to graph the given inequalities. Furthermore, the article also explains the various steps involved to solve a linear inequality in two variables using a graph.

The solution for a linear inequality is always a region bounded by a line on one side. We now know that there are two types of line boundary for the solution region – inclusive and non-inclusive boundaries, for two types of inequalities – strict and slack inequalities.

Showing Inequalities on Number Line

Frequently Asked Questions – Solutions of Linear Inequalities in Two Variables

Q.1. What are linear inequalities in two variables?
Ans: Linear inequalities in two variables is represented as inequality of two algebraic expressions that involve only two variables. For example, \(ax + by \le c\) and \(ax + by > c.\) Here, \(a\) and \(b\) are coefficients, \(x\) and \(y\) are variables, and \(c\) is a constant.

Q.2. How do you plot a linear inequality in two variables?
Ans: For a linear inequality, the solution set is a shaded half-plane. For a \(y > \) inequality, the region above the line is shaded, and for a \(y < \) inequality, the region below is shaded.

Q.3.How do you solve a system of linear inequalities in two variables?
Ans A system of linear inequalities in two variables consists of at least two linear inequalities in the same variables. Steps to solve such a system of inequalities is:
Step 1: Plot the graph of the various inequalities in the system and shade the corresponding solution sets.
Step 2: Identify the intersecting region of all the solution sets in the system
Step 3: The identified region is the solution set of the system of linear inequalities.

Q.4. What is the solution to a system of linear inequality?
Ans: The solution of a system of linear inequalities is the ordered pair that is a solution to all inequalities in the system. The graph of the linear inequality is the graph of all solutions of the system.

Q.5. What is the first step in graphing a linear inequality in two variables?
Ans: The first step to graph a linear inequality is to change the inequality symbol to \( = .\)

Now you are provided with all the necessary information on the solutions of linear equations in two equations and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.