{"id":1395,"date":"2016-10-24T21:30:48","date_gmt":"2016-10-24T21:30:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&p=1395"},"modified":"2017-04-04T16:16:40","modified_gmt":"2017-04-04T16:16:40","slug":"why-it-matters-complex-numbers-and-other-types-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/why-it-matters-complex-numbers-and-other-types-of-functions\/","title":{"raw":"Why It Matters: Equations and Inequalities","rendered":"Why It Matters: Equations and Inequalities"},"content":{"raw":"

How do equations and inequalities help in the real world?<\/h2>\r\n[caption id=\"attachment_3689\" align=\"alignright\" width=\"511\"]\"Seabees U.S. Navy Seabees help build a classroom for a school in Thailand.[\/caption]\r\n\r\n \r\n\r\nYou are in charge of planning and purchasing materials for a new school to be built overseas in a small village. \u00a0The plans for the foundation indicate that it must be rectangular, with a length of exactly 100 feet greater than the width. \u00a0However the total area must be 12,000 sq. ft. \u00a0How long and how wide should the foundation be?\r\n\r\n \r\n\r\nFirst we have to model<\/strong> the problem using mathematics. \u00a0Let [latex]x[\/latex] represent the width (shorter side). \u00a0Then the length must be [latex]x+100[\/latex]. \u00a0Using the area formula for a rectangle, we find:\r\n\r\n \r\n

[latex]Area=\\left(length\\right)\\times\\left(width\\right)=\\left(x+100\\right)\\left(x\\right)=x^2+100x[\/latex]<\/p>\r\n \r\n\r\nNow since the total area must be equal<\/em> to 12,000, we can write:\r\n

[latex]x^2+100x=12,000[\/latex]<\/p>\r\nor equivalently,\r\n

[latex]x^2+100x-12,000=0[\/latex]<\/p>\r\n \r\n\r\nThis is an equation<\/strong>, which is a mathematical statement consisting of two expressions set equal to one another. \u00a0In order to find out the proper length and width of the floor, we would have to solve<\/strong> the equation for the variable x using techniques that you will soon discover.\r\n\r\n \r\n\r\nSuppose the walls of the school will be built from concrete blocks costing $1.20 per block (which includes tax) plus a flat fee of $100 for hauling the blocks to the site. \u00a0Having only a limited budget, you must be very careful to order only what you can afford. \u00a0Suppose the school requires at least 1800 concrete blocks (but having more blocks around for future projects would not be a bad idea). \u00a0Your budgeted amount for concrete is $2,400. \u00a0What is the maximum number of blocks that you can purchase? \u00a0Do you have enough money to buy at least the minimum number of blocks required for the project?\r\n\r\nThe answer to the question above, we must be familiar with inequalities<\/strong>. \u00a0An inequality is a statement involving relations like \u201cless than,\u201d \u201cmore than,\u201d \u201cat least,\u201d or \u201cat most.\u201d \u00a0For example, if the number of blocks to be purchased is n, then there are two inequalities implied by the story.\r\n

[latex]n\\ge1800[\/latex] \u00a0 and \u00a0 [latex](1.20)n+100\\le2400[\/latex]<\/p>\r\nThe first inequality expresses the fact that at least<\/em> 1800 blocks are required. The second inequality states that the total cost of the blocks, together with the hauling fee, cannot exceed the budgeted amount of $2,400.\r\n\r\n \r\n\r\nIn this module, you will learn how to analyze this situation as well as many others that involve building and solving equations and inequalities. \u00a0We\u2019ll revisit our construction site again at the end of the module.\r\n\r\n \r\n

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Learning Objectives<\/h3>\r\nEquation-Solving Techniques\r\n