Linear Inequalities

Linear Inequalities


Introduction

A relation of the type f(x, y) > 0, f(x, y) < 0, f(x, y) 0, or f(x, y) 0 is called an inequality or an inequation. It is also called a constraint or a condition. If f(x, y) is linear in x and y,

then the inequality is called a linear inequality or a linear inequation. The relations ax + b 0 or cx + d 0 are linear inequations in one variable x.


Note:

(i) Equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sign of the inequality.

(ii) Both sides of an inequality can be multiplied (or divided) by the same positive number. But when both sides are multiplied or divided by a negative number, then the sign of the

inequality is reversed.


Solutions of system of linear inequations in one variable

In order to solve a system of linear inequations, we first find the solution set of each of the inequations separately. Then, we find the values of the variable which are common to them or

the intersection of all these sets.


Linear inequations in two variables

Consider the inequation

16 x + 6 y 200. ...(i)


In this example x and y are whole numbers and can not be fractions or negative numbers. In this case we find the pair of values of x and y which make the statement (i) true. The set of

such pairs is the solution set of (i).

To start with, let x = 0 so that

6y 200 or

As such the values of y corresponding to x = 0, can be 0, 1, 2, 3, ...., 33. Hence the solutions of (i) are (0, 0), (0, 1), (0, 2) ,....., (0, 33).

Similarly, the solutions corresponding to x = 1, 2, 3, ..., 12 are

(1, 0), (1, 1), .... , (1, 30),

(2, 0), (2, 1), ...., (2, 28),

We note that the values of x and y cannot be more than 12 and 33 respectively. We also note that some of the pairs namely (5, 20), (8, 12), and (11, 4) satisfy the equation

16x + 6y = 200 which is a part of the given inequation.

Let us now extend the domain of x and y from whole number to the real numbers. Consider the equation 16 x + 6 y = 200

and then draw the straight line represented by it. This line divides the coordinate plane in two half planes.

For (0, 0), (i) yields, 0 200 which is true and hence we conclude that (0, 0) belongs to the half plane represented by (i). Hence the solution of (i) consists of all the points belonging to

the shaded half plane which consists of infinite number of points.




Note:
In order to identify the half plane represented by an inequation, it is sufficient to take any known point (not lying on the line) and check whether it satisfies the given inequation or not. If

it satisfies, then the inequation represent that half plane, containing the known point, otherwise the inequation represents the other half plane.

For a set of linear inequations, in two variables, the solution region may be

(i) a closed region inside a polygon (bounded by straight lines),

(ii) an unbounded region (bounded partly by straight lines),or (iii) empty.