Quadratic Equations and Linear Inequalities, Graphical Representation of Linear Inequalities in One or Two Variables

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Graphical Representation of Linear Inequalities in One or Two Variables

Thus we can generalize as follows:

  • If are real numbers, then is called a linear equalities in two variables and , where as or , and are called linear Inequations in two variables and .

  • The equation is a straight line which divides the plane into two half planes which are represented by and .

For example can be represented by line , in the xy - plane as shown in Fig.

3x + 4y- 12 = 0 in the xy - plane

3x + 4y- 12 = 0 in the Xy - Plane

3x + 4y- 12 = 0 in the xy - plane

The line divides the cordinate plane into two half -plane regions:

  1. Half plane region I above the line

  2. Half plane region II below the line . One of the above region represents the inequality and the other region will be represented by

To identify the half plane represented by Inequation (i), we take any arbitrary point, preferably origin, if it does not lie on . If the point satisfies the Inequation (i), then the half plane in which the arbitrary point lies, is the desired half plane. In this case, taking origin as the arbitrary point we have

i.e.

Thus origin satisfies the inequalities. Now, origin lies in half plane region II. Hence the inequality represents half plane II and the inequality will represent the half plane I.

Before taking more examples, it is important to define the following:

(i) Closed Half Plane: A half plane is said to be closed half plane if all points on the line separating the two half planes are also included in the solution of the inequation.

(ii) An Open Half Plane: A half plane in the plane is said to be an open half plane if the points on the line separting the planes are not included in the half plane.

Example 1:

Show on graph the region represented by the inequalities .

Solution:

The given inequalities is

Let us first take the corresponding linear equation and draw its graph with the help of the following table:

Let Us First Take the Corresponding Linear Equation
Let us first take the corresponding linear equation

Since does not lie on the line , so we can select as the arbitrary point. Since is not true

linear equation x + 2y = 5 in graph

Linear Equation X + 2y = 5 in Graph

linear equation x + 2y = 5 in graph

The desired half plane is one, in which origin does not lie

The desired half plane is the shaded one in Fig.

Example 2:

Represent graphically the inequalities

Solution:

linear equation 3x-12 = 0 in graph

Linear Equation 3x-12 = 0 in Graph

linear equation 3x-12 = 0 in graph

Given inequation is and the corresponding linear equation is or or which is represented by the line on the plane (See Fig.). Taking as the arbitrary point, we can say that and so, half plane II represents the Inequations.

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