document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209290>Question 1209290</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the inequality x(x + 6) > 16 - x^2 + 14x.  Write your answer in interval notation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848154 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848154\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Consider the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x%2B6%29+=+16+-+x%5E2+%2B+14x ALIGN=MIDDLE  ALT=\"x%2A%28x%2B6%29+=+16+-+x%5E2+%2B+14x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's get everything to one side.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x%2B6%29+=+16+-+x%5E2+%2B+14x ALIGN=MIDDLE  ALT=\"x%2A%28x%2B6%29+=+16+-+x%5E2+%2B+14x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x+=+16+-+x%5E2+%2B+14x ALIGN=MIDDLE  ALT=\"x%5E2%2B6x+=+16+-+x%5E2+%2B+14x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x+-+16+%2B+x%5E2+-+14x+=+0 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x+-+16+%2B+x%5E2+-+14x+=+0\"   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+-+8x+-+16+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2+-+8x+-+16+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%5E2+-+4x+-+8%29+=+0 ALIGN=MIDDLE  ALT=\"2%28x%5E2+-+4x+-+8%29+=+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%5E2+-+4x+-+8+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+4x+-+8+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "That last equation is of the format  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where,<BR>\n" );
document.write( "a = 1, b = -4, c = -8<BR>\n" );
document.write( "Plug those into the quadratic formula.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-4%29+%2B-+sqrt%28%28-4%29%5E2+-+4%281%29%28-8%29%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-4%29+%2B-+sqrt%28%28-4%29%5E2+-+4%281%29%28-8%29%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284+%2B-+sqrt%2816%2B32%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%284+%2B-+sqrt%2816%2B32%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284+%2B-+sqrt%2848%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%284+%2B-+sqrt%2848%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284+%2B-+sqrt%2816%2A3%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%284+%2B-+sqrt%2816%2A3%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284+%2B-+sqrt%2816%29%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%284+%2B-+sqrt%2816%29%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284+%2B-+4%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%284+%2B-+4%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282%282+%2B-+2%2Asqrt%283%29%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%282%282+%2B-+2%2Asqrt%283%29%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2+%2B-+2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+2+%2B-+2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2+%2B+2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+2+%2B+2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2+-+2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+2+-+2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5.464 ALIGN=MIDDLE  ALT=\"x+=+5.464\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1.464 ALIGN=MIDDLE  ALT=\"x+=+-1.464\"   DISABLED_style= \"max-width:90%; height:auto;\" > both of which are approximate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Draw a number line with regions A, B, C<BR>\n" );
document.write( "region A = stuff to the left of -1.464<BR>\n" );
document.write( "region B = stuff between -1.464 and 5.464<BR>\n" );
document.write( "region C = stuff to the right of 5.464\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Possible test values for regions A,B,C could be: x = -2, x = 0, x = 6 in that order<BR>\n" );
document.write( "You plug each test value back into the original inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x%2B6%29+%3E+16+-+x%5E2+%2B+14x ALIGN=MIDDLE  ALT=\"x%2A%28x%2B6%29+%3E+16+-+x%5E2+%2B+14x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to see if we get a true statement or not.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If we tried x = -2, then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x%2B6%29+%3E+16+-+x%5E2+%2B+14x ALIGN=MIDDLE  ALT=\"x%2A%28x%2B6%29+%3E+16+-+x%5E2+%2B+14x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2A%28-2%2B6%29+%3E+16+-+%28-2%29%5E2+%2B+14%2A%28-2%29 ALIGN=MIDDLE  ALT=\"-2%2A%28-2%2B6%29+%3E+16+-+%28-2%29%5E2+%2B+14%2A%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8+%3E+-16 ALIGN=MIDDLE  ALT=\"-8+%3E+-16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is true.<BR>\n" );
document.write( "Any x value in region A is in the solution set for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x%2B6%29+%3E+16+-+x%5E2+%2B+14x ALIGN=MIDDLE  ALT=\"x%2A%28x%2B6%29+%3E+16+-+x%5E2+%2B+14x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Region A is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C+2+-+2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"x+%3C+2+-+2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > aka <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity+%3C+x+%3C+2+-+2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"-infinity+%3C+x+%3C+2+-+2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Yielding the interval notation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C3%2C-infinity%2C%22%2C%22%2C2-2sqrt%283%29%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C3%2C-infinity%2C%22%2C%22%2C2-2sqrt%283%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Use curved parenthesis to indicate we do <u>not</u> include the endpoints in the solution set.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If we tried x = 0 then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x%2B6%29+%3E+16+-+x%5E2+%2B+14x ALIGN=MIDDLE  ALT=\"x%2A%28x%2B6%29+%3E+16+-+x%5E2+%2B+14x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2A%280%2B6%29+%3E+16+-+0%5E2+%2B+14%2A0 ALIGN=MIDDLE  ALT=\"0%2A%280%2B6%29+%3E+16+-+0%5E2+%2B+14%2A0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3E+16 ALIGN=MIDDLE  ALT=\"0+%3E+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is false. <BR>\n" );
document.write( "Any x value in region B, ie the region between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2+-+2%2Asqrt%283%29+=+-1.464 ALIGN=MIDDLE  ALT=\"x+=+2+-+2%2Asqrt%283%29+=+-1.464\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2+%2B+2%2Asqrt%283%29+=+5.464 ALIGN=MIDDLE  ALT=\"x+=+2+%2B+2%2Asqrt%283%29+=+5.464\"   DISABLED_style= \"max-width:90%; height:auto;\" > will make the original inequality false.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The last region to test is region C.<BR>\n" );
document.write( "Let's plug in x = 6<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x%2B6%29+%3E+16+-+x%5E2+%2B+14x ALIGN=MIDDLE  ALT=\"x%2A%28x%2B6%29+%3E+16+-+x%5E2+%2B+14x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A%286%2B6%29+%3E+16+-+6%5E2+%2B+14%2A6 ALIGN=MIDDLE  ALT=\"6%2A%286%2B6%29+%3E+16+-+6%5E2+%2B+14%2A6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72+%3E+64 ALIGN=MIDDLE  ALT=\"72+%3E+64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is true<BR>\n" );
document.write( "Therefore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+2+%2B+2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"x+%3E+2+%2B+2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > aka <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%2B+2%2Asqrt%283%29+%3C+x+%3C+infinity ALIGN=MIDDLE  ALT=\"2+%2B+2%2Asqrt%283%29+%3C+x+%3C+infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > translates to the interval notation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C3%2C2+%2B+2sqrt%283%29%2C%22%2C%22%2C+infinity%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C3%2C2+%2B+2sqrt%283%29%2C%22%2C%22%2C+infinity%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We found these two interval notation regions<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C3%2C-infinity%2C%22%2C%22%2C2-2sqrt%283%29%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C3%2C-infinity%2C%22%2C%22%2C2-2sqrt%283%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C3%2C2+%2B+2sqrt%283%29%2C%22%2C%22%2C+infinity%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C3%2C2+%2B+2sqrt%283%29%2C%22%2C%22%2C+infinity%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Glue them together with a set union symbol to end up with this <font color=red>final answer</font><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C3%2C%28matrix%281%2C3%2C-infinity%2C%22%2C%22%2C2-2sqrt%283%29%29%29%2C%22U%22%2C%28matrix%281%2C3%2C2+%2B+2sqrt%283%29%2C%22%2C%22%2C+infinity%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can use a graphing calculator like <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> and <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> to confirm.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );