document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=87680>Question 87680</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am no good at graphing. Please help.<BR>\n" );
document.write( "Find and graph the solution set:<BR>\n" );
document.write( "a) x^3 -16x >0<BR>\n" );
document.write( "b) X^2-5x_< -6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #63598 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=63598\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Look at it this way. Spend 10 times as much time thinking before you even<BR>\n" );
document.write( "try to graph. You're probably too much in a hurry to get it done.<BR>\n" );
document.write( "It's like skiing,skateboarding- you can do better-think more about what you're doing.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+16x+%3E+0 ALIGN=MIDDLE  ALT=\"x%5E3+-+16x+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%5E2+-+16%29+%3E+0 ALIGN=MIDDLE  ALT=\"x%28x%5E2+-+16%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x+%2B+4%29%28x+-+4%29+%3E+0 ALIGN=MIDDLE  ALT=\"x%28x+%2B+4%29%28x+-+4%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now you start thinking<BR>\n" );
document.write( "What would make this inequality NOT true?<BR>\n" );
document.write( "One of the things is if the left side turned out to be 0.<BR>\n" );
document.write( "If any of the 3 factors on the left is 0, then the left is 0.<BR>\n" );
document.write( "Call the left side f(x). So far we have<BR>\n" );
document.write( "x - - - f(x)<BR>\n" );
document.write( "+4 - - - 0<BR>\n" );
document.write( "-4 - - - 0<BR>\n" );
document.write( "0 - - -  0<BR>\n" );
document.write( "These are the roots of f(x).<BR>\n" );
document.write( "What else makes the inequality NOT true?<BR>\n" );
document.write( "You have 3 factors. What does it take to make a negative result?<BR>\n" );
document.write( "A negative result would make the inequality NOT true<BR>\n" );
document.write( "(+)(+)(+) = +<BR>\n" );
document.write( "(-)(-)(+) = +<BR>\n" );
document.write( "(-)(+)(-) = +<BR>\n" );
document.write( "(+)(-)(-) = +<BR>\n" );
document.write( "These combinations make the inequality TRUE -not what we want<BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "(+)(+)(-) = -<BR>\n" );
document.write( "(+)(-)(+) = -<BR>\n" );
document.write( "(-)(+)(+) = -<BR>\n" );
document.write( "(-)(-)(-) = -<BR>\n" );
document.write( "This thinking helps you make SENSE of the graph and is worth the effort<BR>\n" );
document.write( "How can we get any of these last 4 combinations? Try different values<BR>\n" );
document.write( "for x and see<BR>\n" );
document.write( "What if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4+%3C+x+%3C+4 ALIGN=MIDDLE  ALT=\"-4+%3C+x+%3C+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >? Just look at the factors in terms of sign only<BR>\n" );
document.write( "If x is 0, then<BR>\n" );
document.write( "(0)(+)(-) > 0 NOT true<BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C+x+%3C+4 ALIGN=MIDDLE  ALT=\"0+%3C+x+%3C+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then<BR>\n" );
document.write( "(+)(+)(-) > 0 NOT true<BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4+%3C+x+%3C+0 ALIGN=MIDDLE  ALT=\"-4+%3C+x+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. then<BR>\n" );
document.write( "(-)(+)(-) > 0 This is TRUE<BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "So far, we have if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+x+%3C=+4 ALIGN=MIDDLE  ALT=\"0+%3C=+x+%3C=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > the inequality is NOT true<BR>\n" );
document.write( "What about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C+-4 ALIGN=MIDDLE  ALT=\"x+%3C+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > For that I get<BR>\n" );
document.write( "(-)(-)(-) > 0 This is NOT true<BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "The only thing remaining is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E4 ALIGN=MIDDLE  ALT=\"x%3E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I get<BR>\n" );
document.write( "(+)(+)(+) > 0 and this is TRUE<BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "The end result of all this is that the inequality does NOT hold<BR>\n" );
document.write( "for the cases<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+x+%3C=+4 ALIGN=MIDDLE  ALT=\"0+%3C=+x+%3C=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+-4 ALIGN=MIDDLE  ALT=\"x+%3C=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For ALL other cases the inequality holds<BR>\n" );
document.write( "Now I draw the graph and prove it (or not, if I'm wrong)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+700%2C+500%2C+-10%2C+10%2C+-26%2C+26%2C+x%5E3+-+16x%29+ ALIGN=MIDDLE  ALT=\"+graph%28+700%2C+500%2C+-10%2C+10%2C+-26%2C+26%2C+x%5E3+-+16x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The graph seems to verify my statements. <BR>\n" );
document.write( " It's (+) for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E4 ALIGN=MIDDLE  ALT=\"x+%3E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and also for the<BR>\n" );
document.write( "interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4+%3C+x+%3C+0 ALIGN=MIDDLE  ALT=\"-4+%3C+x+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You could reach the same result just by testing a lot of numbers<BR>\n" );
document.write( "and seeing if f(x) is (-), (0), or (+). That's what I do as a last<BR>\n" );
document.write( "resort.<BR>\n" );
document.write( "Hope this helps- you can apply it to your 2nd problem </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );