12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

All this is asking us to do is what we already know from the previous standards, plus one simple step. In fact, this step is fun (as long as you color inside the lines). Students should know how to graph a linear inequality, complete with all the nuts and bolts.

A linear inequality is the same as a linear equation, but instead of an equal sign, we'll have to use the inequality signs (like ≤, ≥, <, and >).

What's all this "half-plane" business? Just mathematical mumbo-jumbo. It means that because we're graphing an inequality and our linear equation is with a different sign now, it'll be shaded above or below the line as part of our solution. That's it.

When dealing with inequalities, your students should ask themselves two questions:

1. Which part of the graph do I shade in?
2. Do I draw a dotted or a solid line?

If the inequality is greater than or greater than or equal to (using either > or ≥), then we shade the upper half of the graph. If the inequality if less than or less than or equal to (using either < or ≤), then we shade the lower half of the graph.

If students are struggling with which half to shade, the simplest way to remove all doubt is to plug in the coordinates of a point that's very obviously on one side of the boundary. If the inequality is true for that point, then we know to shade the "half-plane" containing that point. If it's false, we'll shade in the other half. Make sure to bring your colored pencils.

The line that graphs our linear equation is dashed or dotted if we use greater than or less than (using > or <) in our inequality. That's so we know the line is a boundary, but all the points on it don't satisfy the inequality. The line we'll use is solid if the inequality has a greater than or equal to or less than or equal to (using ≥ or ≤) symbol because the boundary includes possible solutions to our inequality.

Students should understand how to graph not one, but two inequalities. It's just like graphing one inequality, and then graphing another right on top of it. Using the same graph saves trees.

Given a pair of inequalities (such as y < x – 5 and y ≥ x – 6, for instance), we draw them as though they were equations first. We can do this through a computer, a graphing calculator, or by creating a table of values to calculate enough points to get us a straight line.

Time to bust out those colored pencils. Since our first inequality is "less than," this means we must shade below the line. We'll color it red. Then comes the ultimate question: solid or dotted? Well, there's no "equal to" component, so our set of solutions to the inequality does not include the boundary line itself. That means it must be drawn as a dotted line.

For the second inequality, we know that it must be "greater than or equal to," meaning we shade above the line. We'll pick blue. Because of its " equal to" part, we must include the line. It must remain solid.

Red and blue make purple. The overlapping purple area is the solution to our system of inequalities. That means that only within the overlapping area will the values of x and y work for both the inequalities we listed.

Students should know how to graph inequalities, shade in the half-planes, and find the set of solutions for a system of inequalities. If students are struggling, have them plug in coordinates that are on the boundary or very clearly to one side. This will help connect the graph and the inequality, as well as make sense of what's going algebraically and graphically.

Also, make sure they pick colors that go together.