document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205786>Question 1205786</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Brian is studying for his final exams in Chemistry and Algebra. He knows he has less than 22 hours to study, and it will take him at least three times longer to study for Algebra than Chemistry.\r<BR>\n" );
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document.write( "Graph the system on your own graph paper with Chemistry as the independent variable and Algebra as the dependent variable. Then, decide which of the four graphs below shows the correct lines (dotted/solid), and then choose the letter for the region on that graph that should be shaded for the solution. If no regions should be shaded, choose No Solution.\r<BR>\n" );
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document.write( "A B C D E F G H I J K L M N O P No Solution\r<BR>\n" );
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document.write( "Can he spend 14 hours on Chemistry and 9 hours on Algebra?\r<BR>\n" );
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document.write( "Can he spend 15 hours on Chemistry and 9 hours on Algebra? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842840 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=842840\">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 hours to study for Chemistry<BR>\n" );
document.write( "y = number of hours to study for Algebra<BR>\n" );
document.write( "Both are nonnegative so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0 ALIGN=MIDDLE  ALT=\"x%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=0 ALIGN=MIDDLE  ALT=\"y%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "x+y is the total number of hours studied<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By+%3C+22 ALIGN=MIDDLE  ALT=\"x%2By+%3C+22\"   DISABLED_style= \"max-width:90%; height:auto;\" > because he has less than 22 hours to study\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+3x ALIGN=MIDDLE  ALT=\"y+%3E=+3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > since it takes him at least 3 times longer to study for Algebra than Chemistry.<BR>\n" );
document.write( "Example: if he needs x = 4 hours for Chemistry, then he'll need at least 3x = 3*4 = 12 hours for Algebra.\r<BR>\n" );
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document.write( "The system of inequalities is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%3E=0%2Cy%3E=0%2Cx%2By%3C22%2Cy%3E=3x%29 ALIGN=MIDDLE  ALT=\"system%28x%3E=0%2Cy%3E=0%2Cx%2By%3C22%2Cy%3E=3x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The first two inequalities tell us to focus on the upper right quadrant.<BR>\n" );
document.write( "Inside this quadrant, we'll shade below the dashed line x+y = 22 and above the solid line y = 3x. <BR>\n" );
document.write( "The graph is shown at this Desmos link<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/23hbtoqjk1\">https://www.desmos.com/calculator/23hbtoqjk1</a>\r<BR>\n" );
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document.write( "The corner points of this overlapped shaded region are:<BR>\n" );
document.write( "(0,0)<BR>\n" );
document.write( "(0,22)<BR>\n" );
document.write( "(5.5, 16.5)<BR>\n" );
document.write( "Each corner point is the result of intersecting the boundary lines. <BR>\n" );
document.write( "Eg: (5.5, 16.5) is the result of solving the system <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2By=22%2Cy=3x%29 ALIGN=MIDDLE  ALT=\"system%28x%2By=22%2Cy=3x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Be careful to note that points on the dashed boundary are NOT solutions; in contrast, points on the solid boundary are solutions.\r<BR>\n" );
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document.write( "Now onto the question \"Can he spend 14 hours on Chemistry and 9 hours on Algebra?\"<BR>\n" );
document.write( "We have x = 14 and y = 9<BR>\n" );
document.write( "x+y < 22<BR>\n" );
document.write( "14+9 < 22<BR>\n" );
document.write( "23 < 22<BR>\n" );
document.write( "which is <font color=red>false</font><BR>\n" );
document.write( "Therefore he <font color=red>cannot</font> study Chemistry for 14 hours and Algebra for 9 hours.<BR>\n" );
document.write( "The point (14,9) is not inside the shaded region.\r<BR>\n" );
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document.write( "You should find the same conclusion for the second question as well.<BR>\n" );
document.write( "</font> </SPAN>\n" );
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