Linear Inequations in Two Variables

Linear Inequations in Two Variables:

A statement of inequality containing two variables is known as a linear inequation of two variables. Examples of linear inequations of two variables are-

(i) ax + by + c > 0

(ii) ax + by + c < 0

(iii) ax + by + c ≥ 0

(iv) ax + by + c ≤ 0

where a, b and c are real constants.

The set of all ordered pairs of real numbers satisfying a given inequation is called the solution set of the inequation.

As the set of ordered pairs of real numbers corresponds to points of a two-dimensional coordinate plane, the solution set of a given inequation can be represented by points of a coordinate plane. The set of all points satisfying a given inequation is known as the graph of the inequation.

Steps for Drawing the Graph of an Inequation in One Variable:

(i) For the inequation containing the variable x, the graph of x = c is drawn, which is a line (dotted in the case of x > c or x < c to denote that x = c is not included in the solution set and continuous in the case of x ≥ c or x ≤ c) parallel to the Y-axis.

(ii) For the inequations containing the variable y, we draw the graph of y = c (dotted in the case of y > c or y < c and continuous in the case of y ≥ c or y ≤ c) parallel to the X-axis.

(iii) The dotted or continuous line separates the coordinate plane into two regions. An arbitrary point is then taken in any one of the two regions to test whether its coordinates satisfy the given inequation.

(iv) If the coordinates of the point satisfy the given inequation, then the selected region is the desired region to be shaded as all the points in this region represent the desired solution. On the other hand, if the coordinates of the point chosen do not satisfy the given inequation, then the other region would be the desired region to be shaded.

Graphs of Linear Inequations in Two Variables:

To draw the graphs of linear inequations containing two variables, the steps to be followed are-

(i) The inequality sign is replaced by the sign of equality and the equation ax + by + c = 0 is taken which can further be written in the form x/(-c/a) + y/(-c/b) = 1.

(ii) Two points, x = -c/a and y = -c/b are, then, marked on the X and Y-axes respectively suitably choosing the scale so that -c/a and -c/b are both integers (this is usually achieved by taking the L.C.M. of a and b as the unit of scale).

(iii) A line (dotted, if the inequality contains > or < signs and continuous if the inequality shows ≥ or ≤) is then drawn through these two points. This will divide the plane into two halves.

(iv) An arbitrary point is then taken in any one of the two regions to test whether it satisfies the given inequations or not. If it satisfies, then the chosen region is shaded as the solution set. If it does not satisfy, then the other region would be shaded to denote the solution set.

Graphical Solution of Simultaneous Linear Inequations:

For solving simultaneous linear inequations graphically, the two regions satisfying the two inequations are shaded. The common portion satisfies both the inequations and would, naturally, denote the required solution region.