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Two-variable inequalities are relations between two expressions which are not equivalent and they also have two terms that vary. Ax + By > C is one of the forms that these inequalities take. The two-variable inequalities use signs such as ‘≠’ meaning not equal to, ‘?’ meaning that the unknown can take any value from the number that follows the sign, ‘<’ for less than and ‘>’ for greater than. Ozark Furniture makes various types or rockers. Two-variable inequalities are important when this company wants to determine the number of a particular type of rocker that they are to make in order to maximize profits and meet the market demands.
The assignment requires that I write an inequality limiting the number of strip hammocks. First of all, the variable needs to be defined for purpose of clarity. The number of full-size hammocks will be represented with “f” and the number of chair-size hammocks will be represented with “c”. Each full size hammock requires 10 yards of canvas so the expression that will represent the hammock shall be 10f and 5c for the expression representing the chair hammock as it requires 5 yards of canvas. The canvas that can be used does not exceed 2000 yards because that is all the company has available. “Less than or equal to” will be used in the inequality as it is appropriate in explaining that all the canvas can be used. The inequality will thus be:
10f + 5c ≤ 2000
Making c the dependent variable (on the vertical axis) and f is the independent variable (on the horizontal axis) then using the intercepts the inequality can be graphed as an equation. The process involves the following steps:
C-intercept can be found when the value of f is zero.
400> c the intercept, c-intercept, is thus (0, 400).
F-intercept can be found when the value of c is zero.
200>f the intercept, f-intercept, is thus (200, 0).
The curve on the graph will be solid as the inequality is that takes the form of “less than or equal to”. As it moves from left to right it will slope downwards. This problem can only find relevance in the first quadrant of the graph, so the shaded section is from the line towards the origin and stops at the two axes. Graph represents real-world application as it cannot extend into the other quadrants since it is impossible to create negative hammocks. So as to check whether the order can be filled, test points can be used to see if they fall within the relevant section of the graph. Let’s consider points (105, 175) on the graph. The points are inside the shaded region which means the company could fill an order of 105 full size hammocks and 175 chair hammocks. If this was the total number made they would need:
105(10) + 175(5) = 1925 yards of striped canvas and have 75 yards of canvas left over thus this is an order that can be filled by the company.
Considering on the graph point (150, 125) which is outside the shaded region meaning the company will not be able to fill out such an order.
150(10) + 125(5) = 2125 yards of canvas required. They cannot fill the order.
The point (75, 250) is right on the line meaning if the company fills out this order it would be able to deliver if they provided no room for mistake as all the canvas would be needed. The proof for this is: 75(10) + 250(5) = 2000.
If a customer submits an order for 120 full size hammocks and 180 chair hammocks, then on the graph the point (120, 180) is outside of the shaded area therefore the company cannot meet such an order. 120(10) + 180(5) = 1200 + 900 = 2100 showing that the company need an addition of 100 yards of canvas so that they are able to fill this order.
In conclusion, the application of the two-variable inequalities has helped the company in planning and organizing itself when they receive an order. For instance an order for 120 full size hammocks and 180 chair hammocks can be plotted on the graph as (120, 180) and then this plot is used to determine whether or not the company has enough canvas to supply the customer. Inequalities are important in planning within a company as it helps reduce waste and improve customer satisfaction. The procurement department of a company may use the knowledge to organize the supply of raw materials to match the demand of the customers.
Dugopolski, M. (2012). Elementary and intermediate algebra (4th Ed.). New York, NY: McGraw-Hill Publishing.
Krause, E. F. (2012). Taxicab geometry: An adventure in non-Euclidean geometry. Courier Dover Publications.
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