Two-variable Inequality

Shipping is a very important element in modern commerce. And a ship will float if the weight of passengers and crew (x) plus the weight of cargo (y) is less then or equal to the maximum capacity allowable for the ship.

In a middle school, class sizes are limited to 25 students, say. Let x be the number of girls and y the number of boys in a class. The total of girls and boys in a class must not exceed 25.

We have this inequality:

x + y 25

Each pair of values (x, y) that satisfies the condition is said to be a solution of this inequality. One solution is (14, 11), another is (3, 10), yet another (15, 5).

1. To better understand inequalities and x-y pairs, move your mouse in the active x-y plane below. It will show you the point location indicated by the mouse. Which way to move to increase x? decrease x? increase y? decrease y?

   How do you change x while keeping the y value unchanged? or change y while keeping x the same?

2. Now let's see the solutions to this inequality.

   x + y 25

   Move the mouse in the diagram, convince yourself that points on and to the left of the blue line all satisfy the inequality. And those to the right do not.

3. Move the mouse along the blue line. What happens to the sum x + y?

4. Ah, but you say, x and y are numbers of people and they can not be negative or non-integral. That is exactly right. With these additional conditions on the values for x and y, please describe the points in the xy-plane that satisfy these conditions and the inequality.