document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1089814>Question 1089814</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3.<BR>\n" );
document.write( "Solve the following inequality\r<BR>\n" );
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document.write( "42x -13 < 42/x \r<BR>\n" );
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document.write( "•Is the point x = 0 included in the solution set of the inequality?<BR>\n" );
document.write( "Yes or No\r<BR>\n" );
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document.write( "•Are the other finite end points of the intervals included in the solution set?  Yes or No\r<BR>\n" );
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document.write( "•What is the solution set? __________\r<BR>\n" );
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document.write( "Answer in interval notation and Show work.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #704197 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=704197\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> set 42x - 13 equal to 42/x.\r<BR>\n" );
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document.write( "find the intersection points.\r<BR>\n" );
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document.write( "then find the area before, between, and beyond these intersection points that satisfy the criteria.\r<BR>\n" );
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document.write( "start with 42x - 13 = 42/x\r<BR>\n" );
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document.write( "multiply both sides of the equation by x to get:\r<BR>\n" );
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document.write( "x * (42x - 13) = 42\r<BR>\n" );
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document.write( "subtract 42 from both sides of the equation to get:\r<BR>\n" );
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document.write( "x * (42x - 13) - 42 = 0\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "42x^2 - 13x - 42 = 0\r<BR>\n" );
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document.write( "solve this quadratic equation to get:\r<BR>\n" );
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document.write( "x = 1.1666666666667 or x = -0.85714285714286\r<BR>\n" );
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document.write( "42x -13 is equal to 42/x when x = 1.1666666666667 or x = -0.85714285714286\r<BR>\n" );
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document.write( "the intervals you need to test are:\r<BR>\n" );
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document.write( "x < -0.85714285714286\r<BR>\n" );
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document.write( "x > 1.1666666666667\r<BR>\n" );
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document.write( "x between -0.85714285714286 and 1.1666666666667\r<BR>\n" );
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document.write( "x = -1 is less than -.857, so try that first.\r<BR>\n" );
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document.write( "when x = -1:\r<BR>\n" );
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document.write( "42x - 13 is equal to -55.<BR>\n" );
document.write( "42 / x is equal to -42.\r<BR>\n" );
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document.write( "therefore 42x - 13 is less than 42 / x when x < -.857.....\r<BR>\n" );
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document.write( "x = 0 is between x = -.857... and 1.166 ..., so try that next.\r<BR>\n" );
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document.write( "when x = 0:\r<BR>\n" );
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document.write( "42x - 13 is equal to -13.<BR>\n" );
document.write( "42 / x is undefined.\r<BR>\n" );
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document.write( "***** 0 can't be used because 42 / x is undefined when x = 0.\r<BR>\n" );
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document.write( "the interval between -.857... and 1.166.... has to be split into two intervals.\r<BR>\n" );
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document.write( "those intervals would be x > -.857 and < 0, and x > 0 and < 1.166...\r<BR>\n" );
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document.write( "try x = 1 because 1 is greater than 0 and less than x = 1.166....\r<BR>\n" );
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document.write( "when x = 1:\r<BR>\n" );
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document.write( "42x - 13 = 29<BR>\n" );
document.write( "42/x = 42\r<BR>\n" );
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document.write( "42x - 13 is less than 42/x when x is greater than 0 and less then 1.166...\r<BR>\n" );
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document.write( "next try x = -0.5 because -.5 is greater than -.857... and less than 0.\r<BR>\n" );
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document.write( "when x = -.5:\r<BR>\n" );
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document.write( "42x - 13 = -34<BR>\n" );
document.write( "42 / x = -84\r<BR>\n" );
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document.write( "42x - 13 is not less then 42 / x when x is between -.857 and 0\r<BR>\n" );
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document.write( "x = 2 is greater than x = 1.66...., so try that next.\r<BR>\n" );
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document.write( "when x = 2:\r<BR>\n" );
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document.write( "42x - 13 = 71<BR>\n" );
document.write( "42/x = 24\r<BR>\n" );
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document.write( "42x - 13 is not less than 42/x when x is equal to 2.\r<BR>\n" );
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document.write( "this suggests that your solution will be when:\r<BR>\n" );
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document.write( "x is less than -.857...<BR>\n" );
document.write( "x is greater than 0 and less than 1.166....\r<BR>\n" );
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document.write( "this can be seen graphically in the following two graphs.\r<BR>\n" );
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document.write( "the first is the graph of the quadratic equation of y = 42x^2 - 13x - 42\r<BR>\n" );
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document.write( "the second is the graph of y = 42x - 13 and y = 42/x\r<BR>\n" );
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document.write( "remember that if a = b then a - b = 0\r<BR>\n" );
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document.write( "that's the premise behind the creation of the quadratic equation from 42x - 13 = 42/x.\r<BR>\n" );
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document.write( "here's the graphs:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/080601.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/080602.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "they both give you the same intersection points.\r<BR>\n" );
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document.write( "however, the graph of the quadratic equation, while very useful, doesn't show where the original equations have an undefined value.\r<BR>\n" );
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document.write( "the answers to your questions are:\r<BR>\n" );
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document.write( "•Is the point x = 0 included in the solution set of the inequality?<BR>\n" );
document.write( "Yes or No \r<BR>\n" );
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document.write( "the answer is no because the value of 42/x is undefined when x = 0.\r<BR>\n" );
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document.write( "•Are the other finite end points of the intervals included in the solution set? Yes or No \r<BR>\n" );
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document.write( "the finite end points are not included in the solution set because the inequality is less than rather than less than or equal.\r<BR>\n" );
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document.write( "•What is the solution set? __________ \r<BR>\n" );
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document.write( "the solution set is:\r<BR>\n" );
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document.write( "x < -0.85714285714286 or x > 0 and < x < 1.1666666666667\r<BR>\n" );
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document.write( "in interval notation this would be:\r<BR>\n" );
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document.write( "x = ( minus infinity, -0.85714285714286) or (0, 1.1666666666667)\r<BR>\n" );
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document.write( "x cannot be equal to 0 because then 42/x is undefined.\r<BR>\n" );
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document.write( "x cannot be equal to -0.85714285714286 because than 42x - 13 = 42/x\r<BR>\n" );
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document.write( "x cannot be equal to 1.1666666666667 because then 42x - 13 = 42/x\r<BR>\n" );
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document.write( "this last graph shows the areas where 42x - 13 < 42 / x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/080603.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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