Unit 7, Lesson 5
Graphing Linear Inequalities in Two Variables (Part 2)
Integrated Math 1
5.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this lesson, students will be writing linear inequalities that represent constraints in situations and graphing the solution regions. To prepare for that work, students review writing and graphing an equation that represents a situation.
Launch
If needed, explain or show additional images of grass sod and flower beds to students who might be unfamiliar with these landscaping terms.
Activity
NoneA homeowner is making plans to landscape her yard. She plans to hire professionals to install grass sod in some parts of the yard and flower beds in other parts.
Grass sod installation costs \$2 per square foot, and flower bed installation costs \$12 per square foot. Her budget for the project is \$3,000.
- Write an equation that represents the square feet of grass sod, , and the square feet of flower beds, , that she could afford if she used her entire budget.
- On the coordinate plane, sketch a graph that represents your equation. Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite students to share their equation and graph. Discuss with students:
- "In this situation, what does a point on the line mean?" (A combination of square feet of grass sod and square feet of flower beds that the homeowner could have if she spent her entire budget.)
- "What does the vertical intercept of the graph mean?" (The square feet of flower beds she could have if she installs no grass sod.)
- "What does the horizontal intercept of the graph tell us?" (The square feet of grass sod she could install if she installs no flower beds.)
Lesson Synthesis
Display the inequalities and graphs from both activities in the lesson.
Ask students to compare the solution regions and think about what they tell us about the constraints of the situation they represent. Discuss questions such as:
- "How are the solution regions of the two inequalities alike?" (Each region represents all pairs of values that meet a money-related constraint in a situation. They cover a part of the plane. They stop at a line.)
- "How do we know where the boundary line would be for each graph?" (It is the graph of a related equation.)
- "For each inequality, how did we find out which side of the line contains the solutions?" (We tested one or more pairs of values on each side and saw if—when substituted for the variables—they made the inequality true.)
- "In the first situation, some pairs of values that are in the solution region don't make sense in the situation. Can you explain why a pair such as , which is in the solution region, might not be a reasonable option for the homeowner?" (It doesn't quite make sense to cover 1 square foot of the garden with artificial turf and the rest with gravel. The area of the garden might be a lot greater or a lot less than 801 square feet.)
- "In the second situation, we know that fractional values are not meaningful even though they are in the shaded region. Can you think of other reasons that some points in the solution region might not make sense?" (A point like would be in the solution region, but the vendor might not have that many items to sell.)
Some students may point out that it is possible to reason about the side that contains the solutions by reasoning about each context. In these particular examples, this can be done intuitively and correctly. When the boundary line represents the cost of two quantities exactly on budget, smaller values of each quantity would lead to costs that are below the budget. When the boundary line represents a profit of \$100 from selling two kinds of products, a greater number of each product would mean a greater profit.
Emphasize, however, that it is not always the case that the solution region could be reasoned easily or correctly from the context, so it is always a good idea to verify using another method. The optional activity ("Charity Concert") illustrates this point.
Student Lesson Summary
Inequalities in two variables can represent constraints in real-life situations. Graphing their solutions can enable us to solve problems.
Suppose a café is purchasing coffee and tea from a supplier and can spend up to \$1,000. Coffee beans cost \$12 per kilogram, and tea leaves cost \$8 per kilogram.
Buying kilograms of coffee beans and kilograms of tea leaves will therefore cost . To represent the budget constraints, we can write: .
The solution to this inequality is any pair of and that makes the inequality true. In this situation, it is any combination of the kilograms of coffee and tea that the café can order without going over the \$1,000 budget.
We can try different pairs of and to see what combinations satisfy the constraint, but it would be difficult to capture all the possible combinations this way. Instead, we can graph a related equation, , and then find out which region represents all possible solutions.
Here is the graph of that equation.
To determine the solution region, let’s take one point on the line and one point on each side of the line, and see if the pairs of values produce true statements.
A point on the line:
This is true.
Graph of a line, origin O. Horizontal axis, coffee, kilograms, scale is 0 to 140, by 20’s. Vertical axis, tea, kilograms, scale is 0 to 140, by 20’s. Line starts at 0 comma 125 and passes through 30 comma 80. Points 20 comma 40 and 70 comma 100 are identified.
A point below the line:
This is true.
A point above the line:
This is false.
The points on the line and in the region below the line are solutions to the inequality. Let's shade the solution region.
It is easy to read solutions from the graph. For example, without any computation, we can tell that is a solution because it falls in the shaded region. If the café orders 50 kilograms of coffee and 20 kilograms of tea, the cost will be less than \$1,000.