TWO VARIABLE INEQUALITIES

For this assignment, I am going to work with two-variable inequalities and demonstrate the practical application of these inequalities. I am going to use a graph that shows the number of TV’s on the left side and the number of refrigerators on the bottom. Of course this would mean that my x axis is the bottom, and my y axis on the left. The line shows the combination of TV’s and refrigerators that the truck can hold.

The problem I am going to work on is #68 on page 539 . The 18 wheeler truck can hold 330 TV’s with no refrigerators, or 110 refrigerators and no TV’s. When studying this graph, imagine that the triangle region is shaded, and that it represents any given number of coordinates that is in the shaded area is a combination of TV’s and refrigerators that will fit in the truck.

Because there are two points on the graph, I can figure out the slope of the line. So our slope is -3/1

I can now use this slop in point-slope form to write an equation. When we are finished, we will arrive at our linear inequality.

Our point-slope form equation

Replace the slope with-3/1 and (330,0) for the x and y

Add 330 to both sides using the distributive property

Multiply both sides by 1, then add 3x to both sides

Change the equal sign to a less than or equal to symbol. Because the line represents coordinates that can include numbers on that line, as well as anything in the shaded part, we will use a solid line, instead of a dotted line. This is also represented in the symbol having the equal bar

underneath the less than symbol.

Now I am going to answer the two questions that accompany the first part of the assignment.

The first question asks if the truck will hold 71 refrigerators and 118 TV’s. Another way to phrase this is to ask if the coordinates (71,118) fall into the shaded part of the graph. However to be sure, I will plug the ordered pair into my inequality to see if the combination of cargo will actually fit.

331 ? 330Although it is very close, this combination of TV’s and refrigerators will not fit in the truck. Therefore, the statement is false.

My second question asks if the truck will hold 51 refrigerators and 176 TV’s. Will the coordinates (51,176) fall into the shaded region of our graph? I will work this test point just as the last.

329 ? 330This will be a tight fit, but all of the cargo will be able to make it onto the truck.

Now I have to figure out a minimum or maximum allowance of cargo to be found. The Burbank Buy More is going to make an order which will include, at most, 60 refrigerators. What is the maximum number of TV’s that could also be delivered on the same truck? To find the answer, I will plug 60 into the x place of my inequality and solve for y. Is my inequality

replace x with 60

multiply, then subtract 180 from both sides

This is the maximum amount of TV’s that can be shipped with the refrigerators.

For the second scenario, the Burbank Buy More decides to have a television sale so they change their order to include at least 200 TV’s. I need to figure out the maximum number of refrigerators that can be shipped in the same truck. I start with the same inequality. Replace y with 200 and solve for x

Subtract 200 from each side

Now I divide both sides by 3

This means that no more than 43 refrigerators can be shipped with the 200 TV’s.

While graphing inequalities is a way to figure out if a set of coordinates will fall into our shaded, or solution, region. Once a point is found to be in this shaded area, we can use the known slope (in this case it is), to find parallel coordinates that will also work without having to solve the inequality. Just like the examples I showed today, sometimes it is not easy to determine if the ordered pair is a part of the shaded area. These instances require plugging the coordinates into, and solving, the inequality to determine if the amounts will work or not.

Reference

Dugopolski, M. (2012). Elementary and Intermediate Algebra. New York, NY: McGraw-Hill.