document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183959>Question 1183959</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me find the solution set to the inequality\r<BR>\n" );
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document.write( "|1 - 4x^2| - |11x - 12 - 2x^2| <= 1.\r<BR>\n" );
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document.write( "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 #814491 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814491\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "By far the easiest way to find the solution set is with a graphing calculator.  Graph the two functions<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=abs%281-4x%5E2%29-abs%2811x-12-2x%5E2%29 ALIGN=MIDDLE  ALT=\"y=abs%281-4x%5E2%29-abs%2811x-12-2x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and see where the curve is below y=1.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C2%2C-30%2C10%2Cabs%281-4x%5E2%29-abs%2811x-12-2x%5E2%29%2C1%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-10%2C2%2C-30%2C10%2Cabs%281-4x%5E2%29-abs%2811x-12-2x%5E2%29%2C1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Visually, the solution is the interval between about -6.6 and +1.  A graphing calculator gives the approximate interval to several decimal places as (-6.566084, 1.0660844).<br><BR>\n" );
document.write( "An exact algebraic solution requires a lot of work....<br><BR>\n" );
document.write( "The behavior of the function on the left changes when either of the expressions in absolute value changes sign.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-4x%5E2+=+%281-2x%29%281%2B2x%29 ALIGN=MIDDLE  ALT=\"1-4x%5E2+=+%281-2x%29%281%2B2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The behavior changes are x=-1/2 and x=1/2.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%281-4x%5E2%29=1-4x%5E2 ALIGN=MIDDLE  ALT=\"abs%281-4x%5E2%29=1-4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the interval (-1/2,1/2);<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%281-4x%5E2%29=4x%5E2-1 ALIGN=MIDDLE  ALT=\"abs%281-4x%5E2%29=4x%5E2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > everywhere else<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x-12-2x%5E2=-%282x%5E2-11x%2B12%29+=+-1%282x-3%29%28x-4%29 ALIGN=MIDDLE  ALT=\"11x-12-2x%5E2=-%282x%5E2-11x%2B12%29+=+-1%282x-3%29%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The behavior changes at x=3/2 and x=4.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28-%282x%5E2-11x%2B12%29%29=-%282x%5E2-11x%2B12%29 ALIGN=MIDDLE  ALT=\"abs%28-%282x%5E2-11x%2B12%29%29=-%282x%5E2-11x%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the interval (3/2,4);<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28-%282x%5E2-11x%2B12%29%29=%282x%5E2-11x%2B12%29 ALIGN=MIDDLE  ALT=\"abs%28-%282x%5E2-11x%2B12%29%29=%282x%5E2-11x%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > everywhere else<br><BR>\n" );
document.write( "The critical points of the complete function divide the x-axis into 5 intervals on which the analysis needs to be done separately.<br><BR>\n" );
document.write( "(1) (-infinity,-1/2)<BR>\n" );
document.write( "(2) (-1/2,1/2)<BR>\n" );
document.write( "(3) (1/2,3/2)<BR>\n" );
document.write( "(4) (3/2,4)<BR>\n" );
document.write( "(5) (4,infinity)<br><BR>\n" );
document.write( "On intervals (1), (3), and (5),<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%281-4x%5E2%29-abs%2811x-12-2x%5E2%29=%284x%5E2-1%29-%282x%5E2-11x%2B12%29=2x%5E2%2B11x-13 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and the inequality is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B11x-13%3C=1 ALIGN=MIDDLE  ALT=\"2x%5E2%2B11x-13%3C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B11x-14%3C=0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B11x-14%3C=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The quadratic formula gives the zeros of that quadratic expression as<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-11-sqrt%28233%29%29%2F4 ALIGN=MIDDLE  ALT=\"%28-11-sqrt%28233%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = -6.566084<BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-11%2Bsqrt%28233%29%29%2F4 ALIGN=MIDDLE  ALT=\"%28-11%2Bsqrt%28233%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1.066084<br><BR>\n" );
document.write( "Note one of those solutions is in interval (1) and the other is in interval (3), so both are valid solutions; and those solutions agree with what we found with a graphing calculator.<br><BR>\n" );
document.write( "I'll let you do the details if you want; but on intervals (2) and (4) there are no real solutions, so the results we have found are the complete answer.<br><BR>\n" );
document.write( "ANSWER: the solution set to the inequality is from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-11-sqrt%28233%29%29%2F4 ALIGN=MIDDLE  ALT=\"%28-11-sqrt%28233%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-11%2Bsqrt%28233%29%29%2F4 ALIGN=MIDDLE  ALT=\"%28-11%2Bsqrt%28233%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
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