In this activity, students write an equation to represent a situation, and then they write an associated inequality. They 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 . 

When students start with an equation representing a context and use its structure to write a related inequality, they notice and make use of structure (MP7).

The purpose of this discussion is for students to understand that to find the solution to an inequality, it helps to find the solution to the related equation. That value is important to know because it separates numbers that are solutions to the inequality from numbers that are not solutions.

Ask students to share the inequality they wrote to represent the number of subscriptions Andre must sell to reach his goal.  For example, . Explain that to find whether the solution to this inequality is or , we can substitute some values of that are greater than and some that are less than to check. Alternatively, we can think about the context: If Andre wants to make more money, he needs to sell more magazines, not fewer.

In the same way, we can think: If Diego wants to spend less than $35, he needs to spend less for socks, not more. This will help us understand that why is the solution, not .

To promote thinking about a general solving strategy, ask:

• “How does solving the related equation help us solve an inequality? What does the solution to the equation 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’ symbols?”
• “How can we check whether a value is a solution to the inequality?”
   

In this activity, students  interact with contexts in which the direction of inequality is the opposite of what they might expect if they were to solve it like an equation. For example, in the second problem, the original inequality is , but the solution to the inequality is .

Some students might solve the associated equation and then test values of 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.

Since the task requires students to interpret the meaning of their answer in a context, they are reasoning abstractly and quantitatively (MP2). 

The purpose of the discussion is to let students voice their reasoning about the direction of the inequality symbol using 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 be the focus.

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 . The value -10 is a solution to Kiran’s inequality , even though he can’t spend a negative number of dollars. Students can argue that negative values for simply don’t make sense in this context. Some may argue that we should interpret to mean that Kiran deposits or earns $10 every month.