In this activity, students express an inequality that includes the boundary value using only the symbols and . This generates a need for the symbol , which is introduced in the Activity Synthesis. Students also recall the fact that a closed circle is used to graph an inequality that includes the boundary value.

Monitor for students who express the answer to the last question using words or using symbols. The responses to the last question will be used to introduce the new notation.

Students attend to precision (MP6) when they carefully choose the symbols to express the possible values for the height in the last question.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they used the number line to show the possible heights and how they expressed the possible heights using symbols. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “height is greater than 60 or height is equal to 60” already on it and be updated with the more concise phrase “the height is greater than or equal to 60.”)

The purpose of this discussion is to introduce the symbols and . Ask selected students to share their response to the last question. They are likely to write something like “ or .” The and symbols are not enough to capture what we need here with a single mathematical statement. Introduce the new symbols and  that mean “less than or equal to” and “greater than or equal to.” Invite students to use one of these new symbols to revise their answer to the last question. ( or ).

In this activity, students interpret the notation , , , and to evaluate whether different values make an inequality true. It is not expected that students will solve the inequalities generally.

Monitor for students who use the following approaches:

• Substitute a value in for , evaluate the expression, and think about whether the statement is true. For example, “Is -300 less than 75? Yes, so this is true.”
• Draw a number line for each inequality and use it to reason about different values.

In this partner activity, students take turns deciding whether a value makes an inequality true. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Much discussion takes place between partners. The purpose of the discussion is to note the consequences of an inequality using versus , and versus . 

Direct students’ attention to and . Substituting 25 for in each of these inequalities gives , which is false, and , which is true. The key distinction is that and inequalities are considered true when both sides are equal, whereas and  inequalities are considered false when both sides are equal. Emphasize that substituting a value in for , and thinking about whether the resulting inequality is saying something true, is the most direct way to check whether the value is a solution.

• “What was a strategy you used for determining whether a value makes an inequality true?”
• “Were you surprised by or initially incorrect about any of the answers?”