Solving Problems with Inequalities in Two Variables
10 min
Teacher Prep
Required Preparation
Narrative
In this Warm-up, students use graphing technology to graph simple linear inequalities in two variables. They practice adjusting the graphing window until the solution regions become visible and give useful information. Later in the lesson, students will write inequalities that represent constraints in different situations and find the solution sets. The exercises here prepare students to do the latter, using graphing technology.
Launch
Give students access to graphing technology. If using Desmos:
- Explain to students that typing "<= " gives the ≤ symbol and typing ">=" gives the ≥ symbol.
- Remind students that the + and − buttons can be used to zoom in and out of the graphing window, and that the wrench button in the upper-right corner can be used to set the graphing window precisely.
If using other graphing technology available in your classroom:
- Demonstrate how to enter the ≤ and ≥ symbols.
- Remind students how to set a useful graphing window by zooming in or out, and how to set a precise graphing window by specifying the horizontal and vertical boundaries.
- (For technology that takes equations or inequalities in slope-intercept form only.) Remind students that some inequalities might need to be rewritten such that y is isolated before the inequality can be entered into the graphing tool.
Student Task
Use graphing technology to graph the solution region of each inequality and then sketch each graph. Adjust the graphing window as needed to show meaningful information.
y>x
y≥x
y<-8
-x+8≤y
y<10x−200
2x+3y>60
Sample Response
Sample graphs:
y>x
y≥x
y<-8
-x+8≤y
y<10x−200
2x+3y>60
Activity Synthesis (Teacher Notes)
Display the correct solution regions for all to see, and ask students to check their graphs. Discuss any challenges that students may have come across when trying to graph using technology.
Explain to students that they will now use graphing technology to find solutions to some inequalities that represent constraints in situations.
Standards
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- HSA-REI.D.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
25 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Solving Problems with Inequalities in Two Variables
10 min
Teacher Prep
Required Preparation
Narrative
In this Warm-up, students use graphing technology to graph simple linear inequalities in two variables. They practice adjusting the graphing window until the solution regions become visible and give useful information. Later in the lesson, students will write inequalities that represent constraints in different situations and find the solution sets. The exercises here prepare students to do the latter, using graphing technology.
Launch
Give students access to graphing technology. If using Desmos:
- Explain to students that typing "<= " gives the ≤ symbol and typing ">=" gives the ≥ symbol.
- Remind students that the + and − buttons can be used to zoom in and out of the graphing window, and that the wrench button in the upper-right corner can be used to set the graphing window precisely.
If using other graphing technology available in your classroom:
- Demonstrate how to enter the ≤ and ≥ symbols.
- Remind students how to set a useful graphing window by zooming in or out, and how to set a precise graphing window by specifying the horizontal and vertical boundaries.
- (For technology that takes equations or inequalities in slope-intercept form only.) Remind students that some inequalities might need to be rewritten such that y is isolated before the inequality can be entered into the graphing tool.
Student Task
Use graphing technology to graph the solution region of each inequality and then sketch each graph. Adjust the graphing window as needed to show meaningful information.
y>x
y≥x
y<-8
-x+8≤y
y<10x−200
2x+3y>60
Sample Response
Sample graphs:
y>x
y≥x
y<-8
-x+8≤y
y<10x−200
2x+3y>60
Activity Synthesis (Teacher Notes)
Display the correct solution regions for all to see, and ask students to check their graphs. Discuss any challenges that students may have come across when trying to graph using technology.
Explain to students that they will now use graphing technology to find solutions to some inequalities that represent constraints in situations.
Standards
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- HSA-REI.D.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.