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Give students 3 minutes of quiet work time followed by a whole-class discussion. Students are not expected to be able to make correct predictions in the first question. The purpose of this question is to prompt students to think about how negative values in an inequality affect the solution. It also emphasizes that we should not jump to conclusions about solutions without carefully studying what the inequality means.
Here is an inequality:
3
-3
4
-4
4.001
-4.001
Graph all possible solutions to the inequality on the number line:
The purpose of the discussion is to highlight how negatives in the inequality sometimes make it hard to predict what the solutions will be.
Select students to share how their predictions differed from their final solutions.
To illustrate a simple case where solutions go in the opposite direction on the number line, ask how the solutions to are different from the solutions to .
In this activity, students solidify a process for solving inequalities: first solve the associated equation to find the boundary point, then test a value to determine on which side of that boundary the solutions lie. When students apply the reasoning used for the more scaffolded problems to solve new problems, they express regularity in repeated reasoning (MP8).
Monitor for different choices students make in the process of determining solutions. Examples:
Keep students in groups of 2. Give 5–10 minutes of quiet work time, time to share their responses and reasoning with a partner, and follow with a whole-class discussion.
Select work from students with different strategies, such as those described in the Activity Narrative, to share later.
Graph the solutions to on the number line.
Graph the solutions to on the number line.
Solve the inequality , and graph the solutions on the number line.
Solve the inequality , and graph the solutions on the number line.
The goal of this discussion is to highlight that once we have found the boundary value for an inequality, there are a variety of valid options when testing one or more values on either side as solutions.
Display 2–3 approaches from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:
The key takeaway is that any inequality can be solved with the same general procedure, with choices to make along the way:
Here is an inequality: . The solution set for this inequality is all the values that can be used in place of to make the inequality true. Each solution is one value that makes the inequality true.
In order to solve this inequality, we can first solve the related equation to get the solution . That means 2 is the boundary between values of that make the inequality true and values that make the inequality false.
To solve the inequality, we can check numbers greater than 2 and less than 2 and see which ones make the inequality true.
Let’s check a number that is greater than 2: . Replacing with 5 in the inequality, we get or just . This is true, so is a solution. This means that all values greater than 2 make the inequality true. We can represent the solutions as and also represent the solutions on a number line:
Notice that 2 itself is not a solution because it's the value of that makes equal to 18, and so it does not make true.
For confirmation that we found the correct solution, we can also test a value that is less than 2. If we test , we get or just . This is false, so and all values of that are less than 2 are not solutions.
Let's investigate the inequality .
| -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | |
| -7 | -6 | -4 | -1 | 1 |
Graph the solutions to on the number line:
Here is an inequality: .
Complete the table. Does it match your prediction?
| -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | |
Graph the solutions to on the number line:
Here is an inequality: .
Complete the table. Does it match your prediction?
| -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | |
Graph the solutions to on the number line:
Some students may say shows the solutions for the inequality in the first question, since that is the place where the integer value of first surpasses the number -2. Remind these students that there are values between 1 and 2. Ask them whether 1.1 is a solution, for example.
Some students may graph only whole-number (or only integer) solutions. Ask these students to think about whether values in between whole numbers (or integers) are also solutions.