Linear programming and shaded regions

Last time

Before we start: last time, in Linear equations and changing the subject, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

A garden with limits

A gardener has room for at least one tree and at least two shrubs, but no more than seven plants altogether. Each rule is an inequality, and together they allow only some combinations. Plotted on a graph, the allowed combinations form a region. Today you draw that region, read it, and use it to find a best answer.

From an inequality to a region

Every inequality has a boundary line, which you get by replacing the inequality sign with an equals sign. Draw the boundary. If the sign is less than or equal, or greater than or equal, draw a solid line. If the sign is strict, less than or greater than, draw a dashed line, because points on the line are not allowed. The solution is one side of the line, a whole half of the graph.

Which side?

To choose the side, test a point not on the line. Zero, zero is easiest, unless the line goes through it. For x plus y less than or equal to seven: zero plus zero is zero, which is less than seven. So the side containing the origin is allowed. In exams you shade the unwanted side, and leave the region R unshaded. Read the question to see which it wants.

Three lines, one region

Here are the three boundary lines. x equals one is vertical, y equals two is horizontal, and x plus y equals seven slopes down. They form a triangle with corners at one, two, then one, six, and five, two. That triangle is the region R.