Graphical Solution of Linear Inequalities of Two Variables

The line divides the Cartesian plane into two parts. Each part is known as a half plane. A vertical line will divide the plane in left and right half planes and a non-vertical line will divide the plane into lower and upper half planes.

Working rules for solving using graph.[a > 0, b < 0]

Step 1: Draw the graph of using any two points.

are points on the and respectively.

Step 2:
Substitute (0,0) in the given inequation. i.e. 0 > c if this is true (c is negative) shade the region corresponding to the region containing (0, 0). Otherwise, shade the other region which does not include (0, 0).
Let a = 1, b = 2, c = 3 in the example we have taken.
i.e.

The line divides the Cartesian plane into two parts; I above the line and II below the line.
Substitute (0, 0) in the inequality
, which is not true.
The region containing (0, 0) is not the solution region.
i.e. I is the solution region.

Step 3:
Consider points on the line namely (1, 1), (3, 0) and (5, -1).

It is obvious that these points do not satisfy the inequality. If the inequality is in strictly less or strictly more than form, the line is not included in the solution and so it has to be shown as dotted line.

Example 1:
Solve Real Numbers) graphically in two dimensional plane.

Solution:

Substitute
Substitute

The region containing (0, 0) is the solution region of

Note: The line is also included in the solution.

Example 2:
Solve graphically in two-dimensional plane.

Solution:

Let us take the graph of (0, 10) and (-6, 0) are points on the line graph.

(0,0) when substituted in we get 0<30, which is true.
Origin includes the solution region. And the line does not include
Shaded region with dotted line is the solution set (Points on the line excluded).