The easiest way to solve this problem is to use a diagram to show the inequalities. You may find it helpful to sketch one for yourself.

You will find that they define the finite area enclosed by the line x + y = 2 and x = y², which has vertices at (1, 1) and (4, -2).

If we want the largest possible value for x – y or y – x in this area (or on its boundary) we can consider where lines of the form y = x + c intersect with this area.

The least value of c is – 6, as y = x – 6 goes through the vertex (4, -2).

The greatest value of c is 0, as y = x goes through the vertex (1, 1).

Hence the maximum difference between x and y with the given conditions is 6.

Challenge 313: Maximum Difference