Lesson 14

This warm-up highlights the link between an inequality and its associated equation. This will be solidified throughout the lesson as students solve the associated equation and reason in context to determine the direction of inequality. Notice students who use the value -10 as a boundary as they test values to find solutions to the inequalities.

Give students 5 minutes of quiet work time followed by a whole-class discussion. Optionally, provide students with blank number lines for scratch work.

Display two number lines for all to see that each include -10 and some integral values to its left and right. Ask a few students to share their responses to the first two questions, recording their responses on one number line and gauging the class for agreement. Ask a few students to share their responses to the last two questions, recording their responses on the other number line and gauging the class for agreement. 

Highlight the fact that \(\text-x = 10\) and \(2x = \text-20\) have the same solution (-10), but the inequalities \(\text-x > 10\) and \(2x > \text-20\) don't have the same solutions. Select students to share strategies they had for finding solutions. If not mentioned by students, discuss the fact that since -10 makes the sides equal, the neighborhood of values around -10 is a good place to start looking for solutions.

Previously in this unit, students wrote expressions and equations that are similar to the ones in this activity. Here, they are prompted in a scaffolded way to notice that they can express not just that an outcome can be equal to a value, but that an outcome can be at least as much as a value by using the new notation \(\geq\).

Optionally, provide access to blank number lines to use for scratch work.

Arrange students in groups of 2. Allow 10 minutes of quiet work time and partner discussion followed by a whole-class discussion. Depending on the needs of your class, you may decide to ask students to pause after the first question for the whole-class discussion before tackling the second question.

Here is what we want students to understand as a result of this activity:
In order to find the solution to an inequality like \(3n + 25 \geq 68\), we can solve an equation to find the point where \(3n + 25 = 68\). This is the point that separates numbers that are solutions to the inequality from numbers that are not solutions. To find whether the solution to the inequality is \(n \geq 14\frac13\) or \(n \leq 14\frac13\), we can substitute some values of \(n\) that are greater than \(14\frac13\) and some that are less than \(14\frac13\) to check. Alternatively, we can think about the context: If Andre wants to make more money, he needs to sell more magazines, not fewer. If Diego wants to spend less than $35, he needs to spend less for socks, not more. Ask students:

• How does solving the equation help us solve an inequality? What does the solution tell us about solutions to the inequality?
• What are some ways we can determine whether the solution to an inequality should use less than or greater than?
• How can we check whether a value is a solution to the inequality?
• Could Andre sell exactly \(14\frac13\) subscriptions?
• Can Diego pay exactly $3.01 for each pair of socks?
• How can we tell if there are restrictions on the solutions of the inequality, such as only positive numbers or only whole numbers?

The purpose of this activity is for students to interact with contexts in which the direction of inequality is the opposite of what they might expect if they try to solve like they would with an equation. For example, in the second problem, the original inequality is \(9(7-x) \leq 36\), but the solution to the inequality is \(x \geq 3\).

Some students might solve the associated equation and then test values
of \(x\) to determine the direction of inequality. That method will be introduced in more generality in the next lesson. This activity emphasizes thinking about the context in deciding the direction of inequality.

Keep students in the same groups. Give 5–10 minutes of quiet work time and partner discussion followed by a whole-class discussion.

The purpose of the discussion is to let students voice their reasoning about the direction of the inequality by reasoning about the context. Ask students to share their reasons for choosing the direction of inequality in their solutions. Some students may notice that the algebra in both problems involves multiplying or dividing by a negative number. Honor this observation, but again, the goal is not to turn this observation into a rule for students to memorize and follow. Interpreting the meaning of the solution in the context should remain at the forefront.

As students model real-world situations, questions about the interpretation of the mathematical solution should continue to come up in the conversation. For instance, the amount of the granola bar discount cannot be $3.5923, even though this is a solution to the inequality \(x \geq 3\). The value -10 is a solution to Kiran’s inequality, even though he can’t withdraw -10 dollars. Students can argue that negative values for \(x\) simply don’t make sense in this context. Some may argue that we should interpret \(x=\text-10\) to mean that Kiran deposits $10 every month.