two-variable inequality

INSTRUCTOR GUIDANCE EXAMPLE: Week Three Assignment

2-Variable Inequality

Here is an example of a problem very similar to the one in the Week Three Assignment:

Catskills Hammock Company can obtain at most 2000 yards of striped canvas for making

its full size and chair size hammocks. A full size hammock requires 10 yards of canvas

and the chair size requires 5 yards of canvas. Write an inequality that limits the number

of striped hammocks of each type which can be made.

(b) First I must define what variables I will be using in my inequality.

Let f = the number of full size hammocks

Let c = the number of chair size hammocks

Since each full size hammock requires 10 yards of canvas I will use 10f, and since each

chair hammock requires 5 yards of canvas I will use 5c. The total amount of canvas

which can be used is limited to 2000 yards because that is all they can get. Together my

inequality will look like this:

10f + 5c ≤ 2000

(d) If we call f the independent variable (on the horizontal axis) and c the dependent

variable (on the vertical axis) then we can graph the equation using the intercepts.

The f-intercept is found when c = 0:

10f ≤ 2000

f ≤ 200 The f-intercept is (200, 0).

The c-intercept is found when f = 0:

5c ≤ 2000

c ≤ 400 The c-intercept is (0, 400).

Because this is a “less than or equal to” inequality the line will be solid, sloping

downward as it moves from left to right. The region of the graph which is relevant to this

problem is restricted to the first quadrant, so the shaded section is from the line towards

the origin and stops at the two axes.

(e) Consider the point (105, 175) on my graph. It is inside the shaded area which means

the company could fill an order of 105 full size hammocks and 175 chair hammocks. If

they made up this many items they would use

105(10) + 175(5) = 1925 yards of striped canvas and have 75 yards left over.

Consider the point (150, 125) on the graph. It is outside the shaded area which means the

company could not make up enough of both kinds of hammocks to fill this order. They

would run out of canvas before all of them got made.

150(10) + 125(5) = 2125 yards of canvas required. Cannot fill the order.

Consider the point (75, 250). This point is right on the line and this means the company

could fulfill this order exactly without any canvas left over.

75(10) + 250(5) = 2000 Order is filled but no room for mistakes!

(f) If someone calls and submits an order for 120 full size hammocks and 180 chair

hammocks would the company be able to fill this order?

On the graph one can see that this point (120, 180) is outside of the shaded area so

this tells us that no, the company cannot make enough striped hammocks to fill this order.

By plugging in the numbers for each type of hammock and evaluating we can see how

much short of enough canvas the company is.

120(10) + 180(5) = 1200 + 900 = 2100 yards of canvas needed.

The company is 100 yards of canvas short of being able to fill this order.

[Students should remember this example does not demonstrate APA style, nor the

inclusion of introductory or concluding paragraphs.]