document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1152497>Question 1152497</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sporting goods manufacturer allocates at least 1,890 units of time per day to make fishing rods and reels. It takes 7 units of time to make a rod and 21 units of time to make a reel. Write an inequality that describes the possible combinations of the number of units of time devoted to make rods (x) and the number of units of time devoted to make reels (y).\r<BR>\n" );
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document.write( "Give three ordered pair solutions! THnaskk nasdl'; a </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #774512 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=774512\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "x = number of rods made<BR>\n" );
document.write( "y = number of reels made<BR>\n" );
document.write( "Both x and y are non-negative integers (ie whole numbers that cannot be negative).\r<BR>\n" );
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document.write( "\"It takes 7 units of time to make a rod\". If you make x of them, then it takes 7*x units of time to make all x rods.\r<BR>\n" );
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document.write( "It takes \"21 units of time to make a reel\". If you make y reels, then you need 21*y units of time to make all of them. \r<BR>\n" );
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document.write( "Add up those expressions: 7x+21y<BR>\n" );
document.write( "This represents the total amount of time to make x rods and y reels.<BR>\n" );
document.write( "Let T be the total amount of time taken up. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+7x%2B21y ALIGN=MIDDLE  ALT=\"T+=+7x%2B21y\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\"A sporting goods manufacturer allocates at least 1,890 units of time per day to make fishing rods and reels\". The key phrasing \"at least\" means \"that amount or more\". Specifically, \"at least 1890\" means \"1890 or more\"\r<BR>\n" );
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document.write( "Based on the fact that the manufacturer uses up at least 1890 units of time per day, this means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+%3E=+1890 ALIGN=MIDDLE  ALT=\"T+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+7x%2B21y ALIGN=MIDDLE  ALT=\"T+=+7x%2B21y\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this can be plugged into the inequality above to go from this <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+%3E=+1890 ALIGN=MIDDLE  ALT=\"T+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%2B21y+%3E=+1890 ALIGN=MIDDLE  ALT=\"7x%2B21y+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "---------------------------------------------\r<BR>\n" );
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document.write( "There are infinitely many (x,y) solutions to that inequality we found in the last section.\r<BR>\n" );
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document.write( "To generate a solution, we can pick any non-negative integer we want for x. Let's pick x = 0 as that is the smallest.\r<BR>\n" );
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document.write( "Plug in x = 0 and solve for y<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%2B21y+%3E=+1890 ALIGN=MIDDLE  ALT=\"7x%2B21y+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%280%29%2B21y+%3E=+1890 ALIGN=MIDDLE  ALT=\"7%280%29%2B21y+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace x with 0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21y+%3E=+1890 ALIGN=MIDDLE  ALT=\"21y+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+1890%2F21 ALIGN=MIDDLE  ALT=\"y+%3E=+1890%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 21\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+90 ALIGN=MIDDLE  ALT=\"y+%3E=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So if the company made 0 rods, then they'll make at least 90 reels (ie 90 or more reels).\r<BR>\n" );
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document.write( "With x = 0 we have infinitely many y values to pick from as long as y is 90 or larger (and y is an integer as well). <BR>\n" );
document.write( "So we could have (0,90) or (0,91) or (0,92) and so on. We have found 3 ordered pairs, so we can stop here.\r<BR>\n" );
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document.write( "If you want x to be a different value, then pick some other non-negative integer such as x = 1 and repeat the process from before (plug in x = 1, solve for y)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%2B21y+%3E=+1890 ALIGN=MIDDLE  ALT=\"7x%2B21y+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%281%29%2B21y+%3E=+1890 ALIGN=MIDDLE  ALT=\"7%281%29%2B21y+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace x with 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B21y+%3E=+1890 ALIGN=MIDDLE  ALT=\"7%2B21y+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21y+%3E=+1890-7 ALIGN=MIDDLE  ALT=\"21y+%3E=+1890-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 7 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21y+%3E=+1883 ALIGN=MIDDLE  ALT=\"21y+%3E=+1883\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+1883%2F21 ALIGN=MIDDLE  ALT=\"y+%3E=+1883%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 21\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+89.667 ALIGN=MIDDLE  ALT=\"y+%3E=+89.667\"   DISABLED_style= \"max-width:90%; height:auto;\" > This is approximate\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=+90 ALIGN=MIDDLE  ALT=\"y%3E=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" > Round up to the nearest whole number to get 90. This is the smallest y can get.\r<BR>\n" );
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document.write( "Therefore, x = 1 pairs with any integer y value such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+90 ALIGN=MIDDLE  ALT=\"y+%3E=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Three ordered pairs could be (1,90), (1,91) and (1,92).\r<BR>\n" );
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document.write( "The same (more or less) happens with x = 2 as well<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%282%29%2B21y+%3E=+1890 ALIGN=MIDDLE  ALT=\"7%282%29%2B21y+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14%2B21y+%3E=+1890 ALIGN=MIDDLE  ALT=\"14%2B21y+%3E=+1890\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21y+%3E=+1890-14 ALIGN=MIDDLE  ALT=\"21y+%3E=+1890-14\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21y+%3E=+1876 ALIGN=MIDDLE  ALT=\"21y+%3E=+1876\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+1876%2F21 ALIGN=MIDDLE  ALT=\"y+%3E=+1876%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+89.333 ALIGN=MIDDLE  ALT=\"y+%3E=+89.333\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximate\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+90 ALIGN=MIDDLE  ALT=\"y+%3E=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" > Round up to get 90. \r<BR>\n" );
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document.write( "We do not round to 89 even though 89.333 is closer to 89 than it is to 90. This is because y is 89.333 or larger, but since y is an integer, we have to opt for the \"or larger\" part of the inequality sign. Three ordered pairs could be (2,90), (2,91), and (2,92). The only real difference is that x is now 2 instead of 1 or 0.\r<BR>\n" );
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document.write( "I'll let you practice with x = 3 and see what happens with y. Hint: You should get something other than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+90 ALIGN=MIDDLE  ALT=\"y+%3E=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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