for example
A linear inequality in the two variables x and y looks like
ax + by < c ax + by < c ax + by > c ax + by > c
where a, b, and c are constants.
A solution to an inequality is any pair of numbers x and y that satisfy the inequality.
The rules for finding the solution set of a linear inequality are much the same as those for finding the solution to a linear equation.
Add or subtract the same expression to both sides.
Multiply or divide both sides by the same nonzero quantity; if that quantity is negative, then the inequality must be reversed.
Example 1. Determine the solution set of 5x + 2y < 17.
One solution to this is x = 2 and y = 3, because 5(2) + 2(3) = 16, which is indeed less than or equal to 17.
A pair of numbers that does not form a solution is x = 3 and y = 2, because 5(3) + 2(2) = 19, which is not less than or equal to 17. The pair x = 2 and y = 3 isn't the only solution; as a matter of fact, there are infinitely many solutions.
Since we can't write down all possible solutions to a linear inequality, a good way to describe the set of solutions to any linear inequality is by a graph. If the pair of numbers x and y is a solution, then think of this pair as a point in the plane, so the set of all solutions can be thought of as a region in the xy-plane.
To illustrate how to determine this region; first, we solve the inequality for y in terms of x.
5x + 2y < 17
2y < ~5x+ 17
y < -(5/2) x + 17/2
Next, graph the line y = -(5/2) x + 17/2.
The set of points (x,y) that lie on this line is the set of all (x,y) such that y is exactly equal to -(5/2) x + 17/2.
These points make up part of the set of solutions to the inequality, but not all. We see that y can also be less than -(5/2) x + 17/2, so all points below the line would also be solutions. The shaded region in Figure 1 shows the solution set.