The purpose of this activity is for students to practice solving inequalities and checking their solutions, especially those involving negative numbers. Students also practice representing inequalities written in natural language with symbols and number lines, and connecting the solution of an inequality with a number line. The work in this activity will be useful when students solve more complex inequalities in the associated Algebra 1 lesson. As students make decisions about solving and checking inequalities and make connections between situations and graphs, they make sense of problems and persevere in solving them (MP1).

The goal of this discussion is to help students make sense of their solutions.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they checked their solutions. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Here are questions for discussion:

• "After you checked your answers using a value from your solution, did any of your answers change?" (Yes. Initially I thought the solution was , but I used -3 to check my answer and realized it was incorrect. When I substituted -3 for  in the original inequality, I got  which is not true. This reminds me to change the inequality symbol when I multiply or divide by a negative number.)
• "Could the elevator hold 1,300 pounds?" (No. It holds up to 1,200, so this is the largest amount the elevator can hold.)
• "Could the senator’s approval rating be 54%?" (Yes. The situation says the approval rating can be 3 points above and below 53%, and 54% fits that requirement.)
• "How can the number line be useful in writing the solution to an inequality?" (It helps to see which values are included in the solution and which are not. It can also be used to check your answer after you solve an inequality.)
• "On the graph of the minimum years of experience for a candidate, does it make sense to have an upper bound? What examples can you think of that would make this situation have an upper bound?" (Yes, because realistically a person cannot have infinity years of experience and that is what the current graph of the inequality represents. For example, many people retire after a certain age, so it is unusual for someone to have more than 70 or so years of experience, even if the person began working at age 18. Also, because workers rarely live and work beyond age 110, this creates another limit on how many years of experience someone can actually have.)

Students sort different representations of inequalities during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and to make connections (MP2, MP7).

Monitor for different ways groups choose to categorize the representations, but especially for categories that distinguish between greater than and less than statements as well as those that distinguish between strictly greater or less than statements compared with those that can be equal to.

As students work, encourage them to refine their descriptions of their groupings using more precise language and mathematical terms (MP6).

The goal of this discussion is to see different ways of grouping the representations. Select groups to share their categories and how they sorted their representations. Discuss as many different categories as time allows, but ensure that one set of categories distinguishes between greater than and less than statements as well as those that distinguish between strictly greater or less than statements compared with those that can be equal to. Attend to the language that students use to describe their categories, giving them opportunities to describe their groupings more precisely. Highlight the use of terms like “greater than,” “less than,” “equal to,” “positive,” and “negative.”