document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199000>Question 1199000</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve for x the inequality equation  (x+9)/(x+1) -2 > 0\r<BR>\n" );
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document.write( "The correct answer is one of the following.  Which one is correct?\r<BR>\n" );
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document.write( "A) -1 < x < 7\r<BR>\n" );
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document.write( "B) x > 7\r<BR>\n" );
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document.write( "C) -1 < x < 8\r<BR>\n" );
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document.write( "D) x > 8 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832692 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=832692\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "As the tutor @greenestamps has shown<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B9%29%2F%28x%2B1%29+-2+%3E+0 ALIGN=MIDDLE  ALT=\"%28x%2B9%29%2F%28x%2B1%29+-2+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "simplifies to<BR>\n" );
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document.write( "If x = 7, then the numerator is 0.<BR>\n" );
document.write( "If x = -1, then the denominator is 0, so we must avoid this x value.\r<BR>\n" );
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document.write( "Draw a number line with -1 and 7 marked on it.<BR>\n" );
document.write( "Label the following regions or intervals<BR>\n" );
document.write( "P: -infinity < x < -1<BR>\n" );
document.write( "Q: -1 < x < 7<BR>\n" );
document.write( "R: 7 < x < infinity<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C200%2C-4%2C9%2C-2%2C2%2C%0D%0A%0D%0Aline%28-4-2%2C0%2C9%2B2%2C0%29%2C%0D%0Aline%28-4%2C0.1%2C-4%2C-0.1%29%2C%0D%0Aline%28-3%2C0.1%2C-3%2C-0.1%29%2C%0D%0Aline%28-2%2C0.1%2C-2%2C-0.1%29%2C%0D%0Aline%28-1%2C0.1%2C-1%2C-0.1%29%2C%0D%0Aline%280%2C0.1%2C0%2C-0.1%29%2C%0D%0Aline%281%2C0.1%2C1%2C-0.1%29%2C%0D%0Aline%282%2C0.1%2C2%2C-0.1%29%2C%0D%0Aline%283%2C0.1%2C3%2C-0.1%29%2C%0D%0Aline%284%2C0.1%2C4%2C-0.1%29%2C%0D%0Aline%285%2C0.1%2C5%2C-0.1%29%2C%0D%0Aline%286%2C0.1%2C6%2C-0.1%29%2C%0D%0Aline%287%2C0.1%2C7%2C-0.1%29%2C%0D%0Aline%288%2C0.1%2C8%2C-0.1%29%2C%0D%0Aline%289%2C0.1%2C9%2C-0.1%29%2C%0D%0Alocate%28-4-0.2%2C-0.1%2C%22-4%22%29%2C%0D%0Alocate%28-3-0.2%2C-0.1%2C%22-3%22%29%2C%0D%0Alocate%28-2-0.2%2C-0.1%2C%22-2%22%29%2C%0D%0Alocate%28-1-0.2%2C-0.1%2C%22-1%22%29%2C%0D%0Alocate%280-0.2%2C-0.1%2C%220%22%29%2C%0D%0Alocate%281-0.2%2C-0.1%2C%221%22%29%2C%0D%0Alocate%282-0.2%2C-0.1%2C%222%22%29%2C%0D%0Alocate%283-0.2%2C-0.1%2C%223%22%29%2C%0D%0Alocate%284-0.2%2C-0.1%2C%224%22%29%2C%0D%0Alocate%285-0.2%2C-0.1%2C%225%22%29%2C%0D%0Alocate%286-0.2%2C-0.1%2C%226%22%29%2C%0D%0Alocate%287-0.2%2C-0.1%2C%227%22%29%2C%0D%0Alocate%288-0.2%2C-0.1%2C%228%22%29%2C%0D%0Alocate%289-0.2%2C-0.1%2C%229%22%29%2C%0D%0A%0D%0Acircle%28-1%2C0%2C0.25%29%2C%0D%0Acircle%28-1%2C0%2C0.27%29%2C%0D%0A%0D%0Acircle%287%2C0%2C0.25%29%2C%0D%0Acircle%287%2C0%2C0.27%29%2C%0D%0A%0D%0Ared%28line%28-1-0.27%2C0.02%2C-4%2C0.02%29%29%2C%0D%0Ared%28line%28-1-0.27%2C0%2C-4%2C0%29%29%2C%0D%0Ared%28line%28-1-0.27%2C-0.02%2C-4%2C-0.02%29%29%2C%0D%0Ared%28locate%28-2.5%2C0.5%2C%22P%22%29%29%2C%0D%0A%0D%0Ablue%28line%28-1%2B0.27%2C0.02%2C7-0.27%2C0.02%29%29%2C%0D%0Ablue%28line%28-1%2B0.27%2C0%2C7-0.27%2C0%29%29%2C%0D%0Ablue%28line%28-1%2B0.27%2C-0.02%2C7-0.27%2C-0.02%29%29%2C%0D%0Ablue%28locate%283%2C0.5%2C%22Q%22%29%29%2C%0D%0A%0D%0Agreen%28line%287%2B0.27%2C0.02%2C10%2C0.02%29%29%2C%0D%0Agreen%28line%287%2B0.27%2C0%2C10%2C0%29%29%2C%0D%0Agreen%28line%287%2B0.27%2C-0.02%2C10%2C-0.02%29%29%2C%0D%0Agreen%28locate%288.2%2C0.5%2C%22R%22%29%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note the open holes at -1 and 7.\r<BR>\n" );
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document.write( "Let's select an x value from interval P (shown in red above)<BR>\n" );
document.write( "I'll pick x = -2<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-%28-2%29%2B7%29%2F%28-2%2B1%29+%3E+0 ALIGN=MIDDLE  ALT=\"%28-%28-2%29%2B7%29%2F%28-2%2B1%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%29%2F%28-1%29+%3E+0 ALIGN=MIDDLE  ALT=\"%289%29%2F%28-1%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9+%3E+0 ALIGN=MIDDLE  ALT=\"-9+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is false, so x = -2 is not a solution to the inequality.<BR>\n" );
document.write( "By extension any item in interval P will also be non-solutions.<BR>\n" );
document.write( "This rules out interval P from the solution set.\r<BR>\n" );
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document.write( "Now pick something from interval Q. <BR>\n" );
document.write( "I'll go for x = 0<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-0%2B7%29%2F%280%2B1%29+%3E+0 ALIGN=MIDDLE  ALT=\"%28-0%2B7%29%2F%280%2B1%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%29%2F%281%29+%3E+0 ALIGN=MIDDLE  ALT=\"%287%29%2F%281%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7+%3E+0 ALIGN=MIDDLE  ALT=\"7+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is true. <BR>\n" );
document.write( "You should find similar true results for any other value in interval Q.<BR>\n" );
document.write( "So interval Q is part of the answer.\r<BR>\n" );
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document.write( "Lastly, we need to check interval R.<BR>\n" );
document.write( "Let's say we select x = 8<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-8%2B7%29%2F%288%2B1%29+%3E+0 ALIGN=MIDDLE  ALT=\"%28-8%2B7%29%2F%288%2B1%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%2F%289%29+%3E+0 ALIGN=MIDDLE  ALT=\"%28-1%29%2F%289%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.111+%3E+0 ALIGN=MIDDLE  ALT=\"-0.111+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately<BR>\n" );
document.write( "Like with interval P, we get a false statement<BR>\n" );
document.write( "So anything in interval R leads to a false statement.\r<BR>\n" );
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document.write( "Therefore, the only valid solution set is -1 < x < 7<BR>\n" );
document.write( "Which in interval notation would be (-1,7)<BR>\n" );
document.write( "Be sure not to mix this up with ordered pair notation.\r<BR>\n" );
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document.write( "Confirmation can be done using a graph.<BR>\n" );
document.write( "Desmos or GeoGebra are good free graphing tools.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-10%2C10%2C-15%2C15%2C%0D%0Agraph%28400%2C400%2C-10%2C10%2C-15%2C15%2C-100%2C%28x%2B9%29%2F%28x%2B1%29+-2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The green curve represents y = (x+9)/(x+1) -2 aka y = (-x+7)/(x+1)<BR>\n" );
document.write( "This green curve is above the x axis on the interval -1 < x < 7<BR>\n" );
document.write( "It is below the x axis when either x < -1 or when x > 7.<BR>\n" );
document.write( "The x intercept is 7, and there's a vertical asymptote at x = -1.\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "Answer: <font color=red>Choice A) -1 < x < 7</font><BR>\n" );
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