Graphical Solution of Linear Inequations

We know that a 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 Fig.

A point in the Cartesian plane will either lie on a line or will lie in either of the half planes I or II. We shall now examine the relationship, if any, of the points in the plane and the inequalities ax + by < c or ax + by > c. 
Let us consider the line 
ax + by = c, a ≠ 0, b  ≠ 0 ... (1)
There are three possibilities namely: 
(i) ax + by =  c (ii) ax + by > c (iii) ax + by < c. 
In case (i), clearly, all points (x, y) satisfying (i) lie on the line it represents and conversely. Consider case (ii), let us first assume that b > 0. Consider a point P (α,β) on the line ax + by = c, b > 0, so that aα + bβ = c.Take an arbitrary point Q (α , γ) in the half plane II . fig

Now ,  we interpret, γ > β  (Why?) 
or bγ > bβ    or    aα + b γ > aα + bβ (Why?) 
or aα + b γ > c i.e., Q(α,γ ) satisfies the inequality ax + by > c.
Thus, all the points lying in the half plane II above the line ax + by = c satisfies the inequality ax + by > c. Conversely, let (α, β) be a point on line ax + by = c and an arbitrary point Q(α, γ) satisfying 
ax + by > c 
so that    aα + bγ > c 
⇒ aα + b γ > aα + bβ (Why?) 
⇒ γ > β (as b > 0) 
This means that the point (α, γ) lies in the half plane II.
Thus, any point in the half plane II satisfies ax + by > c, and conversely any point satisfying the inequality ax + by > c lies in half plane II. 
In case b < 0, we can similarly prove that any point satisfying ax + by > c lies in the half plane I, and conversely. 
Hence, we deduce that all  points satisfying ax + by > c lies in one of the half planes II or I according as b > 0 or b < 0, and conversely. 
Thus, graph of the inequality ax + by > c will be one of the half plane (called solution region) and represented by shading in the corresponding half plane.