Writing and Solving Inequalities in One Variable

5 min

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. 

Student Task

Kiran 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.

  1. Represent the constraints in the situation mathematically. If you use variables, specify what each one means.
  2. How many pounds of prepared dishes can Kiran buy? Explain or show your reasoning.

Sample Response

  1. Sample response: 5.29p+2(2.49)≤605.29p +2(2.49) \leq 60, or 5.29p+4.98≤605.29p + 4.98 \leq 60 (or equivalent). pp is the pounds of prepared foods Kiran could buy without going over budget.
  2. Up to 10.4 pounds, or p≤10.4p \leq 10.4. Sample reasoning: After removing the cost of sparkling water, Kiran still has $55.02. Dividing that amount by 5.29 gives 10.4.
Activity Synthesis (Teacher Notes)

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.)
Standards
Addressing
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.1·Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A-CED.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. <em>For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.</em>
  • A-CED.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. <em>For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.</em>
  • A-CED.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. <em>For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.</em>
  • A-CED.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. <em>For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.</em>
  • A-CED.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. <em>For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.</em>
  • HSA-CED.A.1·Create equations and inequalities in one variable and use them to solve problems. <span>Include equations arising from linear and quadratic functions, and simple rational and exponential functions.</span>
  • 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. <span>For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.</span>

10 min

20 min