In this Warm-up, students practice writing an inequality to represent a constraint, reasoning about its solutions, and interpreting the solutions. The work here engages students in aspects of mathematical modeling (MP4).

To write an inequality, students need to attend carefully to verbal clues so they can appropriately model the situation. The word "budget," for instance, implies that the exact amount given or any amount less than it meets a certain constraint, without explicitly stating this. When thinking about how many pounds of food Kiran can buy, students should also recognize that the answer involves a range, rather than a single value. 

Make sure students see that one or more inequalities that appropriately model the situation. Then, focus the discussion on the solution set. Discuss questions such as:

• "What strategy did you use to find the number of pounds of dishes Kiran could buy?" 
• "Does Kiran have to buy exactly 10.4 pounds of dishes?" (No.) "Can he buy less? Why or why not?" (Yes. He can buy any amount as long as the cost of the food doesn't exceed $55.02. This means that he can buy up to 10.4 pounds.)
• "What is the minimum amount he could buy?" (He could buy 0 pounds of food, but it wouldn't make sense if his goal is to feed the club members.)

This activity enables students to use inequalities to solve a problem about a situation whose constraints might be unfamiliar and in which multiple quantities are unknown.

To reason about the problem, students need to interpret the descriptions carefully and consider their assumptions about the situation. To make sense of the situation, some students may define additional variables or use diagrams, tables, or other representations. Along the way, they engage in aspects of modeling (MP4).

Monitor for students who use these strategies:

• Start with an empty tank and think about the hours of mowing if there are 0, 1, 2, 3, . . . gallons of gasoline in the tank. (If the tank is empty, Han could mow for 0 hours. If it has 1 gallon, he could mow for (or 2.5) hours. And so on, up to 5 gallons.)
• Start with a full tank and the maximum hours of mowing and work down to an empty tank. (If the tank has 5 gallons, Han could mow hours. If it has 0 gallons, then Han could mow for 0 hours.)
• Write a series of inequalities and equations (for instance, , where is the capacity of the tank in gallons, , and ), and then solve .

Plan to have students present in this order to support moving students from working with individual values to solving mathematical inequalities and equations.

Many students are likely to say that represents all possible hours of mowing. Look for students who also specify that must be positive or who also write  (or ) for the solution set. Ask them to share their thinking with the class during discussion.

This activity highlights some ways to decide whether the solution set to an inequality is greater than or less than a particular boundary value (identified by solving a related equation).

Students are presented with two approaches to solving an inequality. Both characters (Priya and Andre) took similar steps to solve the related equation, which gave them the same boundary value. But they took different paths to decide the direction of the inequality symbol.

One solution path involves testing a value on both sides of the boundary value, and another involves analyzing the structure of the inequality. The former is more familiar given previous work. The latter is less familiar and thus encourages students to make sense of problems and persevere in solving them (MP1). It is also a great opportunity for students to practice looking for and making use of structure (MP7).

Expect students to need some support in reasoning structurally, as it is something that takes time and practice to develop. 

This optional activity gives students additional practice in reasoning about the solutions to inequalities without a context. Students can match the inequalities and solutions in a variety of ways—by testing different values, by solving a related equation and then testing values on either side of that solution, by reasoning about the parts of an inequality or its structure, or by graphing each side of an inequality as a linear function. 

For all of the inequalities, after students find the boundary value by solving a related equation, the range of the solutions can be determined by analyzing the structure of the inequality. Monitor for students who do so, and ask them to share their reasoning during class discussion later.

As students work, also make note of any common challenges or errors so they can be addressed.