# Lesson 21

From One- to Two-Variable Inequalities

These materials, when encountered before Algebra 1, Unit 2, Lesson 21 support success in that lesson.

## 21.1: Describing Regions of the Plane (5 minutes)

### Warm-up

In this activity, students describe points in different regions of the plane. This will prepare students for looking at regions of the plane as inequalities. At this stage, students do not need to describe the regions using inequalities and can describe the regions in other ways (such as positive or negative values rather than $$x > 0$$ or $$x < 0$$). Additionally, students do not need to make a distinction about whether points on the axis are included in the shaded region at this stage. If time is an issue, consider assigning different students to different graphs and then sharing the solutions at the end of the activity.

### Student Facing

For each graph, what do all the ordered pairs in the shaded region have in common?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is for students to begin thinking about how to describe regions of a plane. Select students to share their descriptions of the points in the shaded region.

• "Where would we shade to include all the points with a positive $$y$$-coordinate and negative $$x$$-coordinate?" (We should shade quadrant II of the plane.)
• "What other regions could you describe?" (Sample response: Quadrant III of the plane has a negative $$x$$-coordinate and negative $$y$$-coordinate.)

## 21.2: More or Less (20 minutes)

### Activity

In this activity, students recall how to write solutions to inequalities in one variable, then they begin to move those solutions onto a plane to expand into thinking about inequalities in two variables.

### Student Facing

1. Write at least 3 values for $$x$$ that make the inequality true.
1. $$x < \text{-}2$$
2. $$x+2 > 4$$
3. $$2x-1 \leq 7$$
2. Graph the solution to each inequality on a number line.

3. Using the inequality $$x < \text{-}2$$, write 3 coordinate pairs for which the $$x$$-coordinate makes the inequality true. Use the coordinate plane to plot your 3 points.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to begin to move students thinking from drawing solutions to one-variable inequalities on a number line towards drawing solutions to two-variable inequalities on a coordinate plane.

Select students to share their solutions. Poll the class about their coordinate pairs for the last question. Plot the points on a coordinate plane for all to see. Then ask, "What do you notice about the points we found that are solutions to the inequality? How is it related to the number line you drew for the same inequality?" (It includes all the points to the left of -2 on the plane. It represents the same region as the number line, but shaded above and below the line.)

Here are some other questions for discussion:

• "Why do some of the number lines have open dots and others have closed dots?" (An open dot means the point is not included in the solution. A closed dot means the point is included in the solution.)
• "How do you think we could indicate on the plane whether points like $$(\text{-}2,0)$$ should be included in your shaded region?" (Students do not need to learn the usual convention for this at this point. Students will learn the convention in the associated Algebra 1 lessons.)

## 21.3: Above or Below the Line (15 minutes)

### Activity

In this activity, students begin to see a line that represents an equation as a boundary condition that breaks the plane into two halves. Students graph a line, determine whether a given point is on or off the line using both the equation and its graph, and then choose a coordinate so that a point will be on the line, above the line, or below the line.

### Student Facing

1. Graph the line that represents the equation $$y = 3x-4$$

2. Is the point $$(4,8)$$ on the line?
1. Explain how you know using the graph.
2. Explain how you know using the equation.
3. Use the 3 points $$(5, a), (\text-7,b)$$ and $$(c,20)$$
1. Write values for $$a, b,$$ and $$c$$ so that the points are on the line.
2. Write values for $$a, b,$$ and $$c$$ so that the points are above the line.
3. Write values for $$a, b,$$ and $$c$$ so that the points are below the line.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is for students to recognize that a line divides the plane into three different parts: on the line and to either side of the line. Select students to share their solutions and reasoning.

• "How many choices did you have for $$a$$ so that it was on the line? How many choices did you have for $$a$$ so that it is above the line?" (There is only 1 choice for $$a$$ so that the point $$(5,a)$$ is on the line. There are infinitely many choices for $$a$$ so that it is above the line.)
• "What do all of the choices for $$a$$ have in common so that the point $$(5,a)$$ is below the line?" (All of the possible choices for $$a$$ make the inequality $$a < 11$$ true.)