Unit 7, Lesson 7
Solutions to Systems of Linear Inequalities in Two Variables
Integrated Math 1
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To make a quilt, a quilter is buying fabric in two colors, light and dark. He needs at least 9.5 yards of fabric in total.
The light color costs \$9 a yard. The dark color costs \$13 a yard. The quilter can spend up to \$110 on fabric.
Here are two graphs that represent the two constraints.
AInequality graphed on a coordinate plane, origin O. Horizontal axis, light fabric, yards, scale from 0 to 16, by 4’s. Vertical axis, dark fabric, yards, scale from 0 to 12, by 4’s. Line starts on vertical axis at approximately 9 point 5, goes through about 4 comma 5 point 5 and ends on the horizontal axis at 9 point 5. The region above the line is shaded.
BInequality graphed on a coordinate plane, origin O. Horizontal axis, Light fabric, yards, scale from 0 to 16, by 4’s. Vertical axis, dark fabric, yards, scale from 0 to 12, by 4’s. Line starts on vertical axis at approximately 8 point 5, goes through about 6 comma 4 and ends on the horizontal axis at 12. The region below the line is shaded.
Select all the pairs that satisfy the length constraint.
Select all the pairs that satisfy the cost constraint.
Here are some situations that you have seen before. Answer the questions for one situation.
Club Donations
Concert Tickets
Advertising Packages
Write a system of inequalities to represent the constraints. Specify what each variable represents.
Use technology to graph the inequalities and sketch the solution regions. Include labels and scales for the axes.
Members of a high school math club are doing a scavenger hunt. Three items are hidden in the park, which is a rectangle that measures 50 meters by 20 meters.
Can you find the hidden items? Sketch a graph to show where each item could be hidden.
Clue 1:
Clue 2:
Clue 3:
Clue 4:
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.
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.
The solutions to a system of inequalities are all the values for the variables that make all of the inequalities true.
This graph shows a system of two inequalities. The solutions of the system are all the coordinate pairs that make both inequalities true. On the graph, the solution is shown as all the points in the region where the graphs of the two inequalities overlap.
Two or more inequalities that represent the constraints in the same situation form a system of inequalities.