Lesson 21

In the first activity of the lesson, students consider whether the expression \(2x+3y\) is greater than, less than, or equal to 12 for given \((x,y)\) pairs. This warm-up familiarizes students with the computation and reasoning that they will need later to determine the solution region of a linear inequality in two variables.

Students could reason about the answers by considering the signs and relative sizes of the \(x\)- and \(y\)-values, rather than performing full computations. This is an opportunity to notice and make use of structure (MP7).

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

To help students recall the meaning of a solution to an inequality, ask: "Which pairs, if any, are solutions to the inequality \(2x + 3y \leq 12\)?" Make sure students recognize that both \((\text-1,\text-1)\) and \((6,0)\) are solutions because they make the inequality true.

Earlier in the unit, students saw that a linear inequality in one variable has many solutions, which are represented by all the points on one side of a number on a number line. They also recalled that a linear equation in two variables has many solutions, which are represented by all the points on the graph of the equation. The work in this activity builds on those understandings.

Students see that the solutions to a linear inequality in two variables can be represented by many points on a coordinate plane, and that the set of points are in a region bounded by a line. To represent the set of all points that are solutions, we can shade that region. Students also see that the points that fall on the other side of the boundary line are not solutions to the inequality.

During the activity, students are not yet expected to recognize that the line is the graph of an equation related to the inequality (although some students may notice that). That insight is made explicit in the synthesis and reinforced in the next activity.

Display the inequality \(2x+3y \le 12\) for all to see, along with a blank coordinate grid.

Remind students that earlier, in the warm-up, we saw that \((0,5)\) and \((\text-5, 10)\) are not solutions to this inequality, but \((6,0)\) and \((\text-1,\text-1)\) are solutions. On the coordinate plane, mark each point that is a solution with a dot and each point that is not a solution with an X (or use different colors to mark the points).

Then, ask each student to identify 3 coordinate pairs that are solutions and 3 pairs that are not solutions to the inequality. Encourage students to use some negative values of \(x\) and \(y\), and to find pairs that are different than those chosen by the people seated around them.

After a few minutes, poll the class to collect all ordered pairs that students identified and plot them on the blank coordinate grid (again, using different symbols or different colors for solutions and non-solutions).

When all the points are plotted, the plot might look something like the following. (Blue points represent solutions and yellow X's represent non-solutions.)

Ask students what they notice about the plotted points. Students are not expected to perfectly articulate the idea of a solution region at this point, but they should notice that the solutions are separated from non-solutions by what appears to be an invisible line that slants downwards from left to right.

When choosing their coordinate pairs, some students may have started with the equation \(2x+3y=12\) and plotted some solutions for that equation. If they suggest that the invisible line might be the graph of \(2x+3y=12\), that's great, but it is unnecessary to point this out otherwise.

Arrange students in groups of 3–4. Tell students that their job in this activity is to plot some points that do and do not represent solutions to a few inequalities. If time is limited, consider assigning 1–2 inequalities to each group.

If students don’t know how to begin finding points that are solutions and points that are non-solutions, suggest that they pick any \((x,y)\) pair and then determine if it makes the inequality true. This way, they don’t have to worry about picking an “incorrect” point.

Display four graphs that are representative of students' work for the four inequalities. (A document camera would be very helpful, if available.) In particular, look for examples where a student decided to “shade” all the points on one side of the boundary line.

Invite students to make some observations about the graphs. Discuss questions such as:

• "What do the two sets of points represent?" (Solutions and non-solutions to the inequalities)
• "If we plot a new point somewhere on the coordinate plane, how can we tell if it is or is not a solution?" (There appears to be a line that separates the solutions from non-solutions. If it's on the same side of the line as other solutions, then it is a solution.)
• "What might be a good way to show all the possible solutions to a linear inequality in two variables? Is there a better way other than plotting individual points?" (We can shade the region that contains the points that are solutions.)

Ask students: "How can we tell where exactly the solution region stops and non-solution region starts?" Solicit some ideas from students. If no one predicts that the line is the graph of an equation related to the inequality, remind students that when we solved inequalities in one variable, we used a related equation to help us identify a boundary value (a point on a number line). We can do the same here.

Take the first inequality, \(x \geq y\), as an example. Explain that:

• The solutions to \(x=y\) are all coordinate pairs where \(x\) and \(y\) are equal. We can graph it as line that goes through \((\text-1, \text-1), (0,0), (1,1), (2,2)\), and so on.
• The solutions to \(x \geq y\) are all coordinate pairs where \(x\) equals \(y\) (such as the points previously noted), and where \(x\) is greater than \(y\) (such as \((2,1), (3, 2), (4,2)\)). The latter are all located below the line.
• We can shade the region below the line to represent the solutions to \(x\geq y\).

In the next activity, students will take a closer look at whether the boundary line itself is part of the solution region. For now, it is sufficient that students see that the graph of an equation that is related to each inequality delineates the solution and non-solution regions.

\(x \geq y\)

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\(\text-2y \geq \text-4\)

\(3x<0\)

\(x+y>10\)

In the preceding activity, students looked at the regions that represent solutions and non-solutions to inequalities. They recognized that the boundary between the two regions is the graph of an equation that is related to the inequality. Students did not, however, look closely at whether the boundary line itself is a part of the solution. That investigation is the focus of this activity.

Students reason with algebraic and graphical representations of inequalities in two directions. They first graph the solutions to given inequalities, and later write inequalities whose solutions could be represented by given graphs.

If many students get stuck on graphing or writing inequalities, consider moving fairly quickly through the activity and using the discussion questions in the Lesson Synthesis to help students gain clarity and focus. 

Display the inequalities \(x \ge y\) and \(x > y\) for all to see. Then, ask students to consider whether the following coordinate pairs are solutions to each inequality.

1. \((5,4)\)
2. \((5,4.9)\)
3. \((5,5)\)

Make sure students understand why all three coordinate pairs are solutions to \(x \ge y\), but only \((5,4)\) and \((5,4.9)\) are solutions to \(x > y\). Display two graphs, each representing one of these inequalities.

Ask students to predict which graph represents which inequality. Consider polling the class on their predictions. 

Explain that the solid line is a way to say that all the points on that line (\(x=y\)) are solutions, and the dashed line is a way to say otherwise. (This is similar to how we use solid and open circles to represent the boundary values of a one-variable inequality on a number line.)  

Because the solutions to \(x>y\) do not include coordinates where \(x\) and \(y\) are equal, the graph of \(x=y\) is drawn with a dashed line. The solutions to \(x\geq y\) do include coordinates where \(x\) and \(y\) are equal, so the graph of \(x=y\) is drawn with a solid line. 

Tell students they will now sketch the solutions of some other inequalities and think about whether or not the boundary line is included in the solutions.

Some students may struggle with the second set of questions because they do not recall how to write an equation for a vertical line or a horizontal line. Suggest that they write the coordinates of several points on the line and look for a pattern. For example, some points on the vertical line are \((3,\text-2), (3, 0), (3,4)\), and \((3,6)\). Noticing that the \(x\)-value is always 3, regardless of the \(y\)-value, may help remind students that the equation is \(x=3\).

Select students to share their sketched graphs for the first set of questions and the inequalities they wrote for the second set of questions. Use their work and explanations to help the class synthesize the new ideas in this lesson. 

See Lesson Synthesis for discussion questions.