Linear Inequalities in Two Variables

The linear inequalities in two variables represent the inequalities between two given algebraic expressions in which the two distinct variables are included. In short, in linear inequalities, you use the greater than sign (>), less than sign (<), greater than or equal to sign (≥) and less than or equal to sign (≤) instead of the equal to sign (=). For linear equalities, you use the Equal to (=)  sign or show equalities between any two given expressions or numbers. Consider the following example:


x + 2 = y, x = 3 - y, etc.


However, in the linear inequalities, you use a relation between the two variables. Consider for example:


x + 5 > y, y - 3 < x, and so on.


Linear Inequalities Definition

Any two given real numbers or two algebraic expressions that are associated with the symbols >, <,  ≥ or ≤, form an inequality of the expression. Consider the following examples of numerical inequality:


7 < 11, 19 > 13


The examples of algebraic linear inequalities include:


x + 6 > y, y < 9 - x, x ≥ y > 10


The signs < and > in these expressions represent the strict inequalities here and the signs ≤ and ≥ represent the slack inequalities. For representing the linear inequalities in one variable in a number line, it is a visual representation. It is a convenient and easy way for representing the solutions of the inequality. Now, let us discuss the graph of the linear inequalities in two variables.


Graphical Solution of Linear Inequalities in Two Variables

As you know that a line divides a Cartesian two-dimensional plane into two equal parts that are called as half-planes.


A vertical line divides this plane into the left and the right half planes as shown below:


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Similarly, a non-vertical line divides the plane into the lower and the upper half-planes:


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Any given point in this Cartesian plane would either lie on the line, lie in the half-plane I or lie in the half-plane II.


Now, let us look at the relationship between these points in the plane and inequalities. It would either be ax + by < c or be ax + by > c.


The expression having two variables x and y along with a constant c could have three different possibilities - ax + by > c, or ax + by < c, or ax + by = c,


wherein, a ≠ 0 and b ≠ 0.


Different Types of Linear Inequalities

Let us talk about the different types of inequalities to help you understand better.


  1. Numerical Inequalities

When only numbers are involved in the numerical expression, then it is said to be a numerical inequality.


For example, 10 > 6, 5 < 9


  1. Literal Inequalities

x < 3, y > 4, z < 11 are all the examples of the literal inequalities.


  1. Double Inequalities

3 < 6 < 8 read as 6 less than 8 and greater than 3 is an example of the double inequality.


  1. Strict Inequality

Any mathematical expressions that involve only the signs < or > are called the strict inequalities.


For example,  2x + 3 < 6, and 2x + 3y > 6


  1. Slack Inequality

Any mathematical expressions that involve only the signs ≤ or ≥ are called to be the slack inequalities.


For example,  2x + 3 ≤ 6, and 2x + 3y ≥ 6


In the above-given examples,  you can say that the expression 2x + 3 < 6 is the linear inequality in one variable since x is the only variable that is present in the expression.


Similarly, in the given expression 2x + 3y ≥ 6, you can say that it is the linear inequality in two variables since there are two variables x and y that are present in the expression.


Note that the expression 4x2 + 2x + 5 < 0 is not considered to be an example of a linear inequality in one variable since x has an exponent 2 in the first term. It is known to be a quadratic inequality.


Solved Example of Linear Inequalities with Two Variables

Solve y < 2x + 1 graphically.


Solution


First, you need to find the solution of the equation. Assume that x = 0 first and then assume that  x = 1.


The given expression is y = 2x +1


Assume that x = 0


Doing so, you get, y = 2(0) +1


Hence, y = 1


Now assume that x = 1


Substituting the value of x, you get, y = 2(1) +1


Therefore, y = 3


Hence, the coordinates of the line are (0, 1), (1, 3).


When you join these two points you would get the line of the equation as y = 2x + 1.


Since the sign of inequality is < which means that it is a strict inequality, the boundary line would be a dashed line.


Then check for the solution region after substituting x = 0 and y = 0.


The expression is y < 2x +1


Substituting the values of x and y, you get, 0 < 2(0) +1


This is a true statement and hence, the solution region would be on the half-plane which consists of the (0, 0) coordinates.


As a result, you need to shade the lower half-plane which is below the boundary line.


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FAQ (Frequently Asked Questions)

1. What is a Linear Inequality?

A linear inequality is in the form of an equation with the equal sign replaced by an inequality sign. The solution of a linear inequality is usually a range of different values instead of one specific value. 


For solving an inequality, you use the same procedure as you use while solving the linear equations with just the modification that when an inequality is to be multiplied or divided by a negative number, the inequality is reversed.

2. How to Solve Linear Inequalities with Two Variables?

If a linear function consists of two different variables, then it is said to be a linear inequality in two variables. 

For example, 2x + 3y > 60


In this expression, you can see that there are two variables x and y and is hence a linear inequality in two variables.


The solution of the linear inequality in two variables in the form ax + by > c is said to be an ordered pair (x, y) which gives you a true statement when you put in the values of x and y into the given inequality. Let us solve the linear inequality using the same method that you use while solving the linear equations.


You need to first find the solution by assuming that first x = 0 and then consider y = 0. This way you can find the solution for the given expression.


Consider the following example:


Solve this expression x + 9y < 27.


Solution


The first step is to find the pair of values of x and y to know the solution set.


Assume that x = 0.


Doing so, you get, 0 + 9y < 27


Solving this further, you get, 9y < 27


Hence, y < 3


Now assume that y = 0


Substituting it in the expression, you get, x + 9(0) < 27


Hence, x < 27


Therefore, the solution for the given inequality is (0, 3) and (27, 0).


This statement means that if x = 0 then the value of y could be less than 3, which is, 0, 1, and 2. You will not be including 3 here since the expression says that it is less than three, and not less than or equal to three. Therefore, the solution set for this is (0, 0), (0, 1), and (0, 2).


All of the coordinates of these ordered pairs would be included in the solution part.


Similarly, you can find the solution set by assuming that the value of x as 1, 2, 3, etc.