Lesson 20

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 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 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 these approaches students may use to find the possible number of hours of mowing:

• Start with an empty tank and think about the hours of mowing if there is 0, 1, 2, 3,... gallons of gasoline in the tank. (If the tank is empty, Han could mow 0 hours. If it has 1 gallon, he could mow \(\frac{1}{0.4}\) or 2.5 hours. And so on, up to 5 gallons.)
• Write a series of inequalities and equations (for instance, \(c \leq 5\), where \(c\) is the capacity of the tank in gallons, \(c = 0.4x\), and \(0.4x \leq 5\)) and then solve \(0.4x \leq 5\).
• 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 \(\frac {5}{0.4}\) hours. If it has 0 gallons, then Han could mow 0 hours.)

Many students are likely to say that \(x \leq 12.5\) represents all possible hours of mowing. Look for students who also specify that \(x\) must be positive or who also write \(x\geq 0\) (or \(0\leq x \leq 12.5\)) for the solution set. Ask them to share their thinking with the class during discussion.

For students struggling to express the value of \(x\) as an inequality, suggest they first try reasoning about the question and finding some possible hours of mowing if Han had, say, 1 gallon or 2 gallons of gasoline.

One likely incorrect answer is \(0 \leq x \leq 2\), the result of multiplying gallons by 0.4 gallons per hour. If students make this mistake, ask, “About how long can the mower mow with one gallon of gas?” Then, ask if it is reasonable that the mower mows for 2 hours with 5 gallons of gas.

Select previously identified students to share their responses, in the order as listed in the Activity Narrative. Also invite students who explicitly stated that \(x\) must be positive to explain why they included that lower boundary in their response. 

Emphasize that although it is probably understood from the context that the amount of gasoline in the tank cannot be negative and that the lowest possible number of hours of mowing is 0, it is not apparent from a mathematical statement like \(x\leq12.5\). Writing \(0\leq x \leq 12.5\) (or \(x\geq0\) and \(x\leq 12.5\)) makes this assumption explicit and leaves less room for misinterpretation.

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 of 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. 

Arrange students in groups of 2. Give students a few minutes of quiet time to make sense of what Andre and Priya have done, and then time to discuss their thinking with their partner. Pause for a class discussion before students move to the second set of questions and try to solve inequalities using Andre and Priya's methods.

Select students to share their analyses of Andre and Priya's work. Make sure students can follow Priya's reasoning and understand how Priya decided on \(x<3\) by focusing her comparison on \(4x + 3\) and \(18-x\).

Students who perform procedural steps on the inequality may find incorrect answers. For instance, in the third inequality, they may divide each side by -9 and arrive at the incorrect solution \(x < \text-4\). Encourage these students to check their work by substituting numbers into the original inequality.

Students should recognize that the solution set for each inequality should be the same regardless of the reasoning method used.

Select as many students as time permits to share how they used a strategy similar to Priya's to determine the solution set of each inequality. There may be more than one way to reason structurally about a solution set. Invite students who reason in different ways to share their thinking. Record and display their thinking for all to see. 

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, once 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 could be addressed. 

Select students who used different strategies—especially those who made use of the structure of the inequalities—to share their thinking. If no students found solution sets by thinking about the features of the inequalities, demonstrate the reasoning process with one or two examples. For instance:

• \(6x \leq 3x\): After finding \(x = 0\) as the solution to \(6x=3x\), we can reason that for \(6x\) to be less than \(3x\), \(x\) must include negative numbers, so the solution must be \(x \leq 0\).
• \(\dfrac{4x-1}{3}>\text-1\): After finding \(x = \text- \frac12\) as the solution to the related equation \(\dfrac{4x-1}{3}=\text-1\), we can reason that as \(x\) gets smaller, \(\dfrac{4x-1}{3}\) is also going to get smaller. For that expression to be greater than -1, \(x\) will have be to greater than \(\text-\frac12\). 

If time permits, ask students to choose a different inequality and try reasoning this way about its solution.