Unit 7, Lesson 7
Solutions to Systems of Linear Inequalities in Two Variables
Integrated Math 1
Lesson Synthesis
To help students make connections between systems of inequalities and systems of equations, display the following graphs for all to see.
Ask students:
- "How are the two sets of graphs alike?" (They have the same two lines. They can tell us about the solutions to individual equations or inequalities, as well as the solutions to systems.)
- "How are they different?" (The first set of graphs show two regions that overlap, bounded by dotted lines. The second set shows two intersecting lines and the lines are solid. One set represents the solutions to a system of linear inequalities.)
- "How can we tell the number of solutions from each set of graphs?" (The graphs representing a system of equations show one point of intersection, so there is only one solution. The graphs representing a system of inequalities show one region of overlap, but there are many points in that region. This means that there are many solutions.)
Student Lesson Summary
In this lesson, two linear inequalities in two variables represent the constraints in a situation. Each pair of inequalities forms a system of inequalities.
A solution to a system of inequalities is any pair that makes both inequalities true, or any pair of values that simultaneously meet both constraints in the situation. The solution to the system is often best represented by a region on a graph.
Suppose there are two numbers, and , and there are two things we know about them.
- The value of one number is more than double the value of the other.
- The sum of the two numbers is less than 10.
We can represent these constraints with a system of inequalities.
There are many possible pairs of numbers that meet the first constraint, for example: 1 and 3, or 4 and 9.
The same can be said about the second constraint, for example: 1 and 3, or 2.4 and 7.5.
The pair and meets both constraints, so it is a solution to the system.
The pair and meets the first constraint but not the second ( is a true statement, but is not true.)
Remember that graphing is a great way to show all the possible solutions to an inequality, so let’s graph the solution region for each inequality.
A graph of an inequality on a coordinate plane, origin O. Each axis from negative 10 to 5, by 5’s. Dashed line starts below x axis and right of y axis, goes through negative 2 point 5 comma negative 5, 0 comma 0, and 2 point 5 comma 5. The region above the dashed line is shaded.
A graph of an inequality on a coordinate plane, origin O. Each axis from negative 10 to 5, by 5’s. A dashed line starts on the y axis at 10, goes through 5 comma 5, and ends on x axis at 10. The region below the dashed line is shaded.
Because we are looking for a pair of numbers that meet both constraints or make both inequalities true at the same time, we want to find points that are in the solution regions of both graphs.
To do that, we can graph both inequalities on the same coordinate plane.
The solution set to the system of inequalities is represented by the region where the two graphs overlap.
A graph of two intersecting inequalities on a coordinate plane, origin O. Each axis from negative 10 to 5, by 5’s. The first dashed line starts below x axis and right of y axis, goes through negative 2 point 5 comma negative 5, 0 comma 0, and 2 point 5 comma 5. The region above the dashed line is shaded. Second line at on the y axis at 10, goes through 5 comma 5, end on x axis at 10. The region below the dashed line is shaded.