Activity
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.
Length | Cost |
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Write an inequality to represent the length constraint. Let represent the yards of light fabric and represent the yards of dark fabric.
Compare your answer: You answer may vary, but here is a sample.
Select three pairs that satisfy the length constraint.
, , and .
Write an inequality to represent the cost constraint.
Compare your answer: Your answer may vary, but here is a sample.
Select three the pairs that satisfy the cost constraint.
, , and .
Explain why satisfies the cost constraint but not the length constraint.
Compare your answer: Your answer may vary, but here is a sample.
2 yards of each color costs $44, which is less than $110, so it satisfies this constraint. However, this is only 4 yards altogether, which is not at least 9.5 yards.
Find at least one pair of numbers that satisfies both constraints. Be prepared to explain how you know.
Compare your answer: Your answer may vary, but here is a sample.
, , and ( (any point that is in the intersection of the two inequalities).
What does the pair of numbers represent in this situation?
Compare your answer: Yoru answer may vary, but here is a sample.
means 10 yards of light fabric and 1 yard of dark fabric. This is 11 yards of fabric altogether, which is more than 9.5. It costs $103 because . This is less than the $110 maximum.
Write the two inequalities together as a system of inequalities.
Compare your answer: Your answer may vary, but here is a sample.
Self Check
Additional Resources
Determine Whether an Ordered Pair Is a Solution of a System of Linear Inequalities
The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.
System of Linear Inequalities
Two or more linear inequalities grouped together form a system of linear inequalities.
A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.
To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs that make both inequalities true.
Solutions of A System of Linear Inequalities
Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.
The solution of a system of linear inequalities is shown as a shaded region in the coordinate system that includes all the points whose ordered pairs make the inequalities true.
To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.
Example
Determine whether the ordered pair is a solution to this system:
a.) b.)
a.) Is the ordered pair a solution?
The ordered pair made both inequalities true. Therefore is a solution to this system.
b.) Is the ordered pair a solution?
The ordered pair made one inequality true, but it made the other one false. Therefore, is not a solution to this system.
Try it
Try It: Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
Determine whether the ordered pair is a solution to the system:
a.) b.)
a. Yes.
Here is how to determine if each ordered pair is a solution:
Step 1 - Choose an ordered pair. Step 2 - Substitute into the inequality. Step 3 - Simplify. Step 4 - True or false statement? True True Since the ordered pair makes both inequalities true, it is a solution to the system of inequalities.
b. No.
Here is how to determine if each ordered pair is a solution:
Step 1 - Choose an ordered pair.
Step 2 - Substitute into the inequality.
Step 3 - Simplify.
Step 4 - True or false statement? False True Since the ordered pair does not make a true statement in the first inequality, it is not a solution to the system of inequalities.