Unit 4, Lesson 3
Writing and Solving Inequalities in One Variable
Algebra 1
3.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-CED.A.1
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
HSA-CED.A.3
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Building Toward
Instructional Routines
Aspects of Mathematical ModelingMaterials
NoneActivity Narrative
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.
Launch
NoneActivity
NoneKiran is getting dinner for his drama club on the evening of their final rehearsal. The budget for dinner is \$60.
Kiran plans to buy some prepared dishes from a supermarket. The prepared dishes are sold by the pound, at \$5.29 a pound. He also plans to buy two large bottles of sparkling water at \$2.49 each.
- Represent the constraints in the situation mathematically. If you use variables, specify what each one means.
- How many pounds of prepared dishes can Kiran buy? Explain or show your reasoning.
Student Response
Loading...
Activity Synthesis
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.)
3.2
Activity
Materials
NoneActivity Narrative
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.
3.3
Activity
Instructional Routines
MLR7: Compare and ConnectMaterials
NoneActivity Narrative
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.
Launch
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's and Priya's methods.
Select students to share their analyses of Andre's and Priya's work. Make sure students can follow Priya's reasoning and understand how Priya decided on by focusing her comparison on and .
Representation: Internalize Comprehension. Display or provide copies of Priya’s work and Andre’s work on separate displays. Write Priya's and Andre’s descriptive phrases (as given on the student-facing materials) in contrasting colors/texts, so students can more easily sort and process information. Some students may benefit from time to read and interpret each strategy one at a time.
Supports accessibility for: Visual-Spatial Processing, Organization
Supports accessibility for: Visual-Spatial Processing, Organization
3.4
Activity
Optional
Instructional Routines
NoneMaterials
NoneActivity Narrative
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.
Launch
NoneActivity
NoneLesson Synthesis
Display these inequalities. Remind students that they are examples of inequalities written and solved in the lesson.
and
Next, tell students that they will see some statements about writing and solving inequalities. Their job is to decide whether they agree or disagree with each statement and to be prepared to defend their response. (One way to collect their responses is by asking them to give a discrete hand signal.)
Display these statements—one at a time—for all to see. After students indicate their agreement or disagreement, select one or two students to explain their reasoning. Then invite others who are not convinced by the reasoning to offer a counterexample or an alternative view.
- We can usually represent the constraints in a situation with a single inequality. (Disagree. Sometimes multiple inequalities are needed, depending on what's happening in the situation.)
- The only way to check the solutions to an inequality is to see if they make sense in a situation. (Disagree. We can also check by substituting some values in the solution set back into the inequality to see if they make the inequality a true statement.)
- The only way to find the solutions to an inequality in one variable is by testing different values for the variable and seeing which ones work. (Disagree. We can also reason about the solutions in context, solve a related equation and test a higher and lower value, or use the structure of the inequality.)
- To express the solutions to an inequality in one variable, we always use the same inequality symbol as in the original inequality. (Disagree. There are different ways to express a solution set. The symbol we use would depend on how we reason about the solutions. For example, and both represent the solution set to .)
Make sure students see that there are reasons for disagreeing with each of these statements and that they can articulate some of the reasons.
Student Lesson Summary
Writing and solving inequalities can help us make sense of the constraints in a situation and solve problems. Let's look at an example.
Clare would like to buy a video game system that costs \$130 and to have some extra money for games. She has saved \$48 so far and plans on saving \$5 of her allowance each week. How many weeks, , will it be until she has enough money to buy the system and have some extra money remaining? To represent the constraints, we can write . Let’s reason about the solutions:
- Because Clare has \$48 already and needs to have at least \$130 to afford the game system, she needs to save at least \$82 more.
- If she saves \$5 each week, it will take at least weeks to reach \$82.
- is 16.4. Any time shorter than 16.4 weeks won't allow her to save enough.
- Assuming she saves \$5 at the end of each week (instead of saving smaller amounts throughout a week), it will be at least 17 weeks before she can afford the game system.
We can also solve by writing and solving a related equation to find the boundary value for , and then determine whether the solutions are less than or greater than that value.
- Substituting 16.4 for in the original inequality gives a true statement. (When , we get .)
- Substituting a value greater than 16.4 for also gives a true statement. (When , we get .)
- Substituting a value less than 16.4 for gives a false statement. (When , we get .)
- The solution set is therefore .
Sometimes the structure of an inequality can help us see whether the solutions are less than or greater than a boundary value. For example, to find the solutions to , we can solve the equation , which gives us . Then, instead of testing values on either side of 0, we could reason as follows about the inequality:
- If is a positive value, then would be less than .
- For to be greater than , must include negative values.
- For the solutions to include negative values, they must be less than 0, so the solution set would be .