Pearson Algebra 1 Common Core, 2011
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Pearson Algebra 1 Common Core, 2011 View details
6. Systems of Linear Inequalities
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Exercise 28 Page 404

We are constructing a fence around a rectangular area in our garden and need to determine all the possible dimensions we can have with our limited amount of fencing. Let's use the guiding questions to find all the possible dimensions.

What Variables Will You Use? What Will They Represent?

For the dimensions we will use the variables and

  • of the garden:
  • of the garden:

How Many Inequalities Do You Need to Write?

We will use three inequalities to represent the given scenario. One will show the total restrictions of the perimeter, and one for the restrictions on each dimension. Since we know that there is only feet of fencing that can be used, we know the perimeter must be less than or equal to
We are given two other limitations on the garden. The first one is that we want the to be at least feet. The second is that we want the to be at least feet. We can create the following inequalities.
We now have three inequalities that we can graph as a system.
We can create the boundary line for Equation (I) by isolating the variable in the equation
Solve for
We can now graph the boundary line. Since the inequality is less than or equal to we will use a solid line. When graphing we keep in mind that neither nor can take negative values.
Diagram showing the graph of w=-l+75
Now we can determine where to shade using a test point. We will use the point for simplicity.
Since the point makes the inequality true, we will shade below to include it in the solution.
Graph showing the solutions to Inequality (I) and a test point
We will follow the same process to graph the next two inequalities. The second inequality, will be a horizontal line. We will use the same test point to determine where we will shade.
Since made the inequality false, we will shade away from the test point.
Diagram showing the solutions to Inequality (I) and Inequality (II) overlapping each other and a test point
Finally, we will now graph the third inequality, This inequality will be a vertical line at One last time, we will use the same test point to determine where we will shade.
Since made the inequality false, we will shade away from the test point.
Diagram showing the solutions to each of the three inequalities overlapping each other and a test point

The dimensions that satisfy all of the limitations are located within the overlapping shaded region created by all three inequalities.

Diagram showing the solution set to the three inequalities