Solutions of inequalities of two variables

A linear inequality in two variables x and y is of the form: 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 number x and y that satisfies the inequality.

Example 2

Determine the solution set of 5x + 2y ≤ 17

Solution

One solution to 5x + 2y < 17 is x =2 and y = 3 because 5(2) + 2(3) = 16, which is indeed less than 17. But the pair x = 2 and y = 3 is not the only solution. As a matter of fact, there are infinitely many solutions. If the pairs 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 x –y plane.

Hence, to illustrate how to determine this region, first express y in terms of x in the inequality.
3x + 2y ≤ 17
2y ≤ -5x + 17
Y ≤ -5/2 x + 17/2
When x = 0, y = 8.5; when y = 0, x = 3 (show in a graph)

The shaded Region is the solution set.