document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=741468>Question 741468</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find k such that for every real x we have (1 + kx)/(1 + x^2) < k\r<BR>\n" );
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document.write( "The answer is k &#8712; (4/3,&#8734;). I want to know how to get this answer without plugging in random values. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #452691 by </B><A HREF=/tutors/aboutme.mpl?userid=tommyt3rd><B>tommyt3rd(5050)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=tommyt3rd><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=tommyt3rd><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=452691\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Assuming k is any real number...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281+%2B+kx%29%2F%281+%2B+x%5E2%29+%3C+k ALIGN=MIDDLE  ALT=\"%281+%2B+kx%29%2F%281+%2B+x%5E2%29+%3C+k\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "leads to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281+%2B+kx%29%2F%281+%2B+x%5E2%29+-+k+%3C+0 ALIGN=MIDDLE  ALT=\"%281+%2B+kx%29%2F%281+%2B+x%5E2%29+-+k+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bkx%29%2F%281%2Bx%5E2%29-k%28%281%2Bx%5E2%29%2F%281%2Bx%5E2%29%29+%3C+0 ALIGN=MIDDLE  ALT=\"%281%2Bkx%29%2F%281%2Bx%5E2%29-k%28%281%2Bx%5E2%29%2F%281%2Bx%5E2%29%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and combining them we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2Bkx%29-k%281%2Bx%5E2%29%29%2F%281%2Bx%5E2%29+%3C+0 ALIGN=MIDDLE  ALT=\"%28%281%2Bkx%29-k%281%2Bx%5E2%29%29%2F%281%2Bx%5E2%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "after simplifying the numerator...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bkx-k-kx%5E2%29%2F%281%2Bx%5E2%29+%3C+0 ALIGN=MIDDLE  ALT=\"%281%2Bkx-k-kx%5E2%29%2F%281%2Bx%5E2%29+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bx%5E2 ALIGN=MIDDLE  ALT=\"1%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is always positive, so to determine when the sign of the expression will be less than zero we only need to look at the numerator:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-kx%5E2%2Bkx%2B%281-k%29 ALIGN=MIDDLE  ALT=\"-kx%5E2%2Bkx%2B%281-k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "using the quadratic formula to find x...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-k+%2B-+sqrt%28+k%5E2-4%2A%28-k%29%2A%281-k%29+%29%29%2F%282%2A%28-k%29%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-k+%2B-+sqrt%28+k%5E2-4%2A%28-k%29%2A%281-k%29+%29%29%2F%282%2A%28-k%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-k+%2B-+sqrt%28+k%5E2-4%2A%28-k%2Bk%5E2%29%29%29%2F%28-2k%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-k+%2B-+sqrt%28+k%5E2-4%2A%28-k%2Bk%5E2%29%29%29%2F%28-2k%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-k+%2B-+sqrt%28+k%284-3k%29+%29%29%2F%28-2k%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-k+%2B-+sqrt%28+k%284-3k%29+%29%29%2F%28-2k%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which only has a positive real value when\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-3k%3C=0 ALIGN=MIDDLE  ALT=\"4-3k%3C=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3C=0 ALIGN=MIDDLE  ALT=\"k%3C=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-3k%3E=0 ALIGN=MIDDLE  ALT=\"4-3k%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3E=0 ALIGN=MIDDLE  ALT=\"k%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So we notice x is only real on the interval \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=k%3C=4%2F3%5D ALIGN=MIDDLE  ALT=\"0%3C=k%3C=4%2F3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This interval gives us boundary conditions to test. After testing (using a graphing utility) we see only the values k>4/3 satisfies the inequality for ALL values of x<BR>\n" );
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