4b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

The flip side of finding equations is turning word problems into inequalities. These guys are a little more loosey-goosey than equations (since they'll have multiple solutions), but they'll still follow most of the same rules students use for modeling equations.

Emphasize to students how important it is to transform a word problem into an inequality before trying to solve it. It's like a detective taking notes when she's on a big important case. She might be able to keep all her facts and deductions in her head, but it's way easier to have everything laid out visually in a case file. (That's why TV detectives always carry those little black notebooks.)

Students should be on the lookout for phrases like "at least," "no more than," "maximum," "minimum," and "at the most" when converting English into inequalities. Money problems involving a budget should also set off their inequality alarms. If Kieran has a budget of $225 and he wants to buy 8 sweaters at x dollars apiece and 25 one-dollar candy bars, students should be able to translate that into 8x + 25 ≤ 225 and figure out how much he can spend on each sweater.

We're talking single-variable inequalities here, so they'll also need to know how to graph these solutions on a number line. In our example with Kieran, we end up with x ≤ 25, which means he can afford to spend $25 on each sweater. That'll give us every value on the number line to the left of (and including) 25. But the context of the problem also tells us x has to be positive, since Kieran can't pay negative dollars for a sweater, no matter how ugly it is.

Cardigans, on the other hand, are a whole different story.