document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=415111>Question 415111</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I can work no more than 36 hours per week at my two jobs. Housecleaning pays $6 per hour, and my  sales job pays $12 per hour.  I need to earn at least $302 per week to cover my expenses.  I need to write a system of inequalities that shows the various numbers of hours I can work at each job. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #291158 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=291158\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x represent the number of hours housecleaning.<BR>\n" );
document.write( "let y represent the number of hours selling.\r<BR>\n" );
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document.write( "your hours equation is x + y <= 36\r<BR>\n" );
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document.write( "your revenue equation is 6x + 12y >= 302\r<BR>\n" );
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document.write( "you have to satisfy both equations simultaneously.\r<BR>\n" );
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document.write( "Since you have 2 inequalities working against each other, the solution is very difficult to show, and you will have very numerous possibilities.\r<BR>\n" );
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document.write( "You pretty much have to look at the total hours you want to work and then solve the problem for those total hours.\r<BR>\n" );
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document.write( "You cannot work less than 25.16666667 hours, and all of those hours have to be sales hours.\r<BR>\n" );
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document.write( "That will give you a revenue of $302 per week.\r<BR>\n" );
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document.write( "It goes up from there.\r<BR>\n" );
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document.write( "For every 1 hour of sales that you don't do, you have to add 2 hours of housecleaning in order to maintain the minimum.\r<BR>\n" );
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document.write( "With 24.16666667 hours of sales, you will need 2 hours of housecleaning.<BR>\n" );
document.write( "With 22.16666667 hours of sales, you will need 4 hours of housecleaning, etc.\r<BR>\n" );
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document.write( "To solve this problem in any meaningful way without going crazy, you have to pick the number of hours you want to work and then make your equation.\r<BR>\n" );
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document.write( "Example:\r<BR>\n" );
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document.write( "Your revenue equation will always be 6x + 12y >= 302\r<BR>\n" );
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document.write( "When x + y = 36, you can then substitute 36-x for y and solve the equation.\r<BR>\n" );
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document.write( "When x + y = 30, you can then substitute 30 - x for y and solve the equation.\r<BR>\n" );
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document.write( "Your break point will be when x + y = 25.16666667.\r<BR>\n" );
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document.write( "When that happens, you set y = 25.16666667 - x and you solve your revenue equation as follows:\r<BR>\n" );
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document.write( "6x + 12y <= 302\r<BR>\n" );
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document.write( "substitute 25.16666667 - x for y to get:\r<BR>\n" );
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document.write( "6x + 12 *(25.16666667 - x) >= 302\r<BR>\n" );
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document.write( "Simplify to get:\r<BR>\n" );
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document.write( "6x + 302 - 12x >= 302\r<BR>\n" );
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document.write( "combine like terms and subtract 302 from both sides of the equation to get:\r<BR>\n" );
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document.write( "-6x >= 0\r<BR>\n" );
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document.write( "multiply both sides of the equation by -1 to get:\r<BR>\n" );
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document.write( "6x <= 0\r<BR>\n" );
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document.write( "divide both sides of the equation by 6 to get:\r<BR>\n" );
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document.write( "x <= 0\r<BR>\n" );
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document.write( "since x can't be less than 0, the only solution is that x = 0.\r<BR>\n" );
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document.write( "So........\r<BR>\n" );
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document.write( "Your total hours have to be greater than or equal to 25.16666667 and less than or equal to 36.\r<BR>\n" );
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document.write( "Within that window, you have to pick the total hours you want to work and then solve the equation for that number of hours.\r<BR>\n" );
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document.write( "Assume you want to work 30 hours.\r<BR>\n" );
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document.write( "The equations are:\r<BR>\n" );
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document.write( "x + y = 30<BR>\n" );
document.write( "This leads to y = 30-x\r<BR>\n" );
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document.write( "6x + 12y >= 302<BR>\n" );
document.write( "Substitute 30-x for y to get:<BR>\n" );
document.write( "6x + 12*(30-x) >= 302<BR>\n" );
document.write( "solve the equation to get:<BR>\n" );
document.write( "x <= 9.66666667\r<BR>\n" );
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document.write( "If you work 30 hours per week, x has to be less than or equal to 9.6666667 hours.\r<BR>\n" );
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document.write( "When you work 9.6666667 hours at x, then you will be working 20.33333333 hours at y and your revenue will be equal to $302.\r<BR>\n" );
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document.write( "If you work 10 hours at x, you will make 10*6 + 20*12 = $300 (not enough).<BR>\n" );
document.write( "If your work 9 hours at x, you will make 9*6 + 21*12 = $306 (more than enough).\r<BR>\n" );
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document.write( "Bottom Line:\r<BR>\n" );
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document.write( "Total hours worked have to be greater than or equal to 25.16666667 and less than or equal to 36.\r<BR>\n" );
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document.write( "Within that window, pick the total hours you want to work and then the equations will tell you how many hours of each you have to work.\r<BR>\n" );
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document.write( "The equations you will work with are:\r<BR>\n" );
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document.write( "x + y = z, where 25.16666667 <= z <= 36<BR>\n" );
document.write( "6x + 12y >= 302\r<BR>\n" );
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document.write( "Then solve for y = z - x and substitute in the second equation to solve for:\r<BR>\n" );
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document.write( "6x + 12*(z-x) >= 302\r<BR>\n" );
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document.write( "I don't know any simpler way to do this.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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