document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=989123>Question 989123</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please tell the way to solve this : x^2-2x > |2x-1|. Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #609554 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=609554\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "To keep yourself from getting all balled up with absolute value bars, intersections and unions of solution sets, and sundry other bumps in this particular road, the best thing to do is to get a visual idea of what the answer is going to look like before you start working out the details.  Hence, graph the two sides of your inequality on the same set of coordinate axes.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=solver91311&illustration_name=ParabolaOverAbsValLinear.jpg\">.\r<BR>\n" );
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document.write( "The magenta line is the graph of the quadratic function and the red line is the graph of the absolute value function.\r<BR>\n" );
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document.write( "We want to find the intervals where the quadratic function is larger than the absolute value function.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^2\ -\ 2x\ >\ |2x\ -\ 1|\">\r<BR>\n" );
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document.write( "Recalling the definition of absolute value, we can create two inequalties:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^2\ -\ 2x\ >\ 2x\ -\ 1\">\r<BR>\n" );
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document.write( "And\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^2\ -\ 2x\ >\ -2x\ +\ 1\">\r<BR>\n" );
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document.write( "Let's begin with the second one and add 2x to both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^2\ >\ 1\">\r<BR>\n" );
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document.write( "Noting that the zeros of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\ -\ 1\ =\ 0\"> are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \pm 1\">,  we can see that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> must either be less than -1 or greater than 1.  But if you look back at the graph, you can see that 1 is an extraneous root since nowhere in the vicinity of 1 is the quadratic function larger than the absolute value function.  Hence, for THIS HALF of the problem, the appropriate interval is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (-\infty,-1)\">.  Note that the interval is open on both ends because the original inequality is strictly greater than.\r<BR>\n" );
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document.write( "Looking at the other half of the problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^2\ -\ 2x\ >\ 2x\ -\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^2\ -\ 4x\ +\ 1\ >\ 0\">\r<BR>\n" );
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document.write( "I'll leave it as an exercise for you to verify that the zeros of the corresponding quadratic equation are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2\ \pm\ \sqrt{3}\">.\r<BR>\n" );
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document.write( "Again, we have an extraneous root.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2\ -\ \sqrt{3}\"> is a number between 0 and 1 where the quadratic function is everywhere less than the absolute value function.  However, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2\ +\ \sqrt{3}\"> is the point where the two functions intersect and after which the quadratic begins to be the larger valued function.  Hence, the appropriate interval for this half of the problem is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (2\ +\ \sqrt{3},\infty)\">.  Again, the \"strictly greater than\" relation in the original inequality demands an open interval.\r<BR>\n" );
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document.write( "Taken together the solution set is the union of the two intervals:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left\{x\ \in\ \mathbb{R}\ :\ x\ \in\ (-\infty,-1)\ \small\cup\ \LARGE(2+\sqrt{3},\infty)\right\}\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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