Running Head: TWO-VARIABLE INEQUALITY
Running head should use a shortened version of the title if the title is long! All capital
letters for the title and the words Running and Head should be capitalized as well.
Two-Variable Inequality (full title; centered horizontally & vertically)
First Name Last Name
MAT 221
Dr. XXXXXXX XXXXXXXX
Date
1
TWO-VARIABLE INEQUALITY
2
Two-Variable Inequality
Be sure to have a centered title on page 1 of your papers!!
[The introductory paragraph must be written by each individual student and the content will vary
depending on what the student decides to focus on in the general information of the topic. YOUR
INTRODUCTION SHOULD CONNECT MATH CONCEPTS AND REAL-WORLD
APPLICATIONS. DO NOT INCLUDE THE DIRECTIONS IN THE INTRO! The following
paragraph is not an introduction to the paper but rather the beginning of the assignment.]
The objective for the first part of the assignment is to write an inequality limiting the
number of strip hammocks. As explained in the text,
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. (Dugopolski, 2012, p. x) (A real page
number would be included if this came from the text.)
To begin, the variable must be defined for clarity. The number of full-size hammocks will be
represented with “f” and the number of chair-size hammocks will be represented with “c”.
Since each full size hammock requires 10 yards of canvas, the expression representing
the multiplication will be 10f; and, since each chair hammock requires 5 yards of canvas, 5c will
be used. The total amount of canvas that can be used is limited to 2000 yards because that is all
the company has available. Since all of the canvas can be used, 2000 must be included in the
solution set so “less than or equal to” will be used in the inequality. As a result, the inequality
demonstrating the relationship between the variables and total canvas is:
10f + 5c ≤ 2000
If f is the independent variable (on the horizontal axis) and c the dependent variable (on
the vertical axis) then the inequality can be graphed the equation using the intercepts. This
TWO-VARIABLE INEQUALITY
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process involves substituting a 0 for a variable and then solving for the other using equation
steps.
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 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. The graph cannot extend into the other quadrants because negative hammocks
are impossible to create so the graph represents real-world application.
To determine if orders can be filled, test points can be used to see if they fall within the
relevant section of the graph. Consider the point (105, 175) on the 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. This means the
order can be filled.
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.
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Consider the point (75, 250). This point is right on the line and this means the company
could fill this order exactly without any canvas left over. There would be no room for mistakes in
this order. 75(10) + 250(5) = 2000
If someone calls and submits an order for 120 full size hammocks and 180 chair
hammocks, the company would need to determine if the order could be filled. On the graph the
point (120, 180) is outside of the shaded area so the company cannot make enough striped
hammocks to fill this order. By substituting in the numbers for each type of hammock and
evaluating it is easy to 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. [This is NOT the conclusion. It is the last portion of the assignment.]
[The conclusion paragraph must be written by each individual student and the content will vary
depending on what the student decides to include in their summary. DO NOT INCLUDE
PERSONAL NARRATIVE LIKE “I LEARNED…” OR “WE CAN DO….”. BE SURE TO
SUMMARIZE WHY THE CONCEPT(S) ARE IMPORTANT AND HOW THEY CAN BE
CONNECTED TO OTHER CONCEPTS.]
TWO-VARIABLE INEQUALITY
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Reference
Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY:
McGraw-Hill Publishing.
Use the word ‘Reference’ or ‘References’ as the title. THERE IS NO COLON OR
UNDERLINE.
Textbook should ALWAYS be included in every assignment! Be sure to use appropriate
indentation (hanging), font (Arial or Times New Roman), and size (12).
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