Two-Variable Inequalities (title required on first line) Continuing last week’s topic of functions and relationships between variables, this week’s work examines a practical application of two-variable inequalities. As the name implies, there are independent and dependent variables, as well as graphic representations of the solutions. Because of the inequality, the graphs and solutions demonstrate a range of possible answers that would work in the given situation. This problem is similar to #68 on page 539 (Dugopolski, 2012) for purposes of demonstrating the math needed for this writing assignment.
A shipping container can carry maximum of 125 sofas and no recliners, or maximum of no sofas and 275 recliners. Study the graph and write an equation to fit the line. Pretend the triangle region is shaded in and change the equation to an inequality describing this region. The diagram is showing the sofas on the x axis and the recliners on the y axis. There are two points on the graph, (275, 0) and (0, 125), so we can compute the slope of this line. The slope is Running Header: TWO-VARIABLE 3 The point-slope form of a linear equation to write the equation itself can now be used. These are the steps we take to arrive at our linear inequality.
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Start with the point-slope form. Substitute the slope for m and (275, 0) for the x and y. Use distributive property and then add 275 to both sides. Multiply both sides by 5. Add 11x to both sides and change to less than or equal to symbol. The graph has a solid line rather than a dotted line indicating that points on the line itself are part of the solution set. This will be true anytime the inequality symbol has the equal to bar. There are two more questions to answer in this section: 1. Will the shipping container hold 80 sofas and 200 recliners? In other words is the point (80, 200) in the solution region.
This is called a test point and it is used to make sure the shading will fall on correct side of the line. To determine the actual solution, the ordered pair must be substituted into the inequality created earlier. If a true statement is made, the combination of sofas and recliners will work. 11x + 5y < 1375 11(80) + 5(200) ? 1375 880 + 1000 ? 1375 1880 < 375 This is a false statement so the container will not hold the amount given. 2. Will the shipping container hold 68 sofas and 125 recliners or will (68, 125) be in the solution region? This is another test point and will be worked the same way.
Running Header: TWO-VARIABLE 4 11x + 5y < 1375 11(68) + 5(125) ? 1375 748 + 625 ? 1375 1373 ? 1375 Yes, is certainly is, so this shipment would be possible. The next problem requires a minimum and/or maximum range to be found. The Burbank Buy More store is going to make an order, which will include at most 75 sofas. What is the maximum number of recliners which could also be delivered in the same shipping container? Describe the restrictions this would add to the existing graph. It is assumes they will get the maximum number of sofas and plug 75 into the linear inequality and solve for y 11x + 5y < 1375
11(75) + 5y ? 1375 Substitute given value and multiply. 825+ 5y ? 1375 Subtract 825 on both sides. 5y ? 550 Divide both sides by 5. y ? 110 The maximum number of recliners that could also be shipped in the same container is 110. The graph would now have a horizontal line at y = 110 and a vertical line at x = 75, and the rectangular shape would be shaded. The only possible shipment amounts would fall inside that rectangle. The next day, the Burbank Buy More decides they will have a recliner sale so they change their order to include at least 170 recliners.
What is the maximum number of sofas which could also be delivered in the same truck? Describe the restrictions this would add to the original graph. Running Header: TWO-VARIABLE 5 11x + 5y < 1375 11x + 5(170) ? 1375 Substitute given value and multiply 11x + 850 ? 1375 Subtract 850 on both sides. 11x ? 525 Divide both sides by 11. x ? 47 (rounded because partial pieces of furniture are not realistic) If 170 recliners are shipped then the maximum possible sofas in the same shipping container would be 47. The graph would now need to have a horizontal line at y = 170 and the enclosed triangular shape above that line would be shaded.
The possible shipment amounts with the minimum number of recliners would fall inside that triangle. Conclusion paragraph would go here. Remember to include 4-5 sentences to make a complete paragraph. Running Header: TWO-VARIABLE 6 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. Text should ALWAYS be included in every assignment! Be sure to use appropriate indentation (hanging), font (Arial or Times New Roman), and size (12).