document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=833746>Question 833746</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I solve this inequality and put it in interval notation? If you can provide step by step on how to simplify the fractions, that would be great! Thanks. \r<BR>\n" );
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document.write( "Solve the inequality. (Hint: First rewrite the inequality by setting one side equal to 0.)\r<BR>\n" );
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document.write( "3/x &#8722; 1 &#8805; 4/x + 5\r<BR>\n" );
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document.write( "So far I have 3/x-1 - 4/x+5 &#8805; 0\r<BR>\n" );
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document.write( "The options for the answers are: \r<BR>\n" );
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document.write( "(&#8722;&#8734;, &#8722;5) &#8746; (5, &#8734;)<BR>\n" );
document.write( "(&#8722;&#8734;, &#8722;5) &#8746; (1, 19]<BR>\n" );
document.write( "    <BR>\n" );
document.write( "(&#8722;&#8734;, &#8722;1) &#8746; (5, 19]<BR>\n" );
document.write( "[&#8722;5, 1) &#8746; (1, 19]<BR>\n" );
document.write( "(&#8722;&#8734;, 19]<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #502730 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=502730\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F%28x-1%29%3E=4%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"3%2F%28x-1%29%3E=4%2F%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F%28x-1%29-4%2F%28x%2B5%29%3E=0 ALIGN=MIDDLE  ALT=\"3%2F%28x-1%29-4%2F%28x%2B5%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We need a common denominator, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%2A%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x-1%29%2A%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is our best bet.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x%2B5%29%2F%28%28x-1%29%28x%2B5%29%29-4%28x-1%29%2F%28%28x%2B5%29%28x-1%29%29%3E=0 ALIGN=MIDDLE  ALT=\"3%28x%2B5%29%2F%28%28x-1%29%28x%2B5%29%29-4%28x-1%29%2F%28%28x%2B5%29%28x-1%29%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=%283%28x%2B5%29-4%28x-1%29%29%2F%28%28x%2B5%29%28x-1%29%29%3E=0 ALIGN=MIDDLE  ALT=\"%283%28x%2B5%29-4%28x-1%29%29%2F%28%28x%2B5%29%28x-1%29%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=%283x%2B15-4x%2B4%29%2F%28%28x%2B5%29%28x-1%29%29%3E=0 ALIGN=MIDDLE  ALT=\"%283x%2B15-4x%2B4%29%2F%28%28x%2B5%29%28x-1%29%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=%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29%3E=0 ALIGN=MIDDLE  ALT=\"%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29=0 ALIGN=MIDDLE  ALT=\"%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=19 ALIGN=MIDDLE  ALT=\"x=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=19 ALIGN=MIDDLE  ALT=\"x=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > is part of the solution.<BR>\n" );
document.write( "The expression on the left is undefined for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(its value does not exist at those points),<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > are not part of the solution.<BR>\n" );
document.write( "The conclusions above eliminate a few options, but we have to work on the problem a little longer.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The expression on the left, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29 ALIGN=MIDDLE  ALT=\"%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "changes sign (from positive to negative values or vice versa)<BR>\n" );
document.write( "at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=19 ALIGN=MIDDLE  ALT=\"x=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Those are the only values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the sign of that expression changes.<BR>\n" );
document.write( "Where is it positive? We know that<BR>\n" );
document.write( "for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29=19%2F%285%2A%28-1%29%29=19%2F%28-5%29=-19%2F5%3C0 ALIGN=MIDDLE  ALT=\"%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29=19%2F%285%2A%28-1%29%29=19%2F%28-5%29=-19%2F5%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%281000%2C200%2C-14%2C26%2C-0.2%2C0.8%2C+grid%280%29%2C%0D%0Ared%28line%28-5%2C-1%2C-5%2C1%29%29%2Cred%28line%281%2C-1%2C1%2C1%29%29%2C%0D%0Agreen%28line%2819%2C-1%2C19%2C1%29%29%2Clocate%28-4%2C0.6%2C%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29%3C0%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So that expression is negative between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and positive for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-5 ALIGN=MIDDLE  ALT=\"x%3C-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , after changing sign at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It also changes to positive to the right of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but further up it changes sign again at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=19 ALIGN=MIDDLE  ALT=\"x=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It is positive between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=19 ALIGN=MIDDLE  ALT=\"x=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=19 ALIGN=MIDDLE  ALT=\"x=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > it is zero, and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E19 ALIGN=MIDDLE  ALT=\"x%3E19\"   DISABLED_style= \"max-width:90%; height:auto;\" > it is negative.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%281000%2C200%2C-14%2C26%2C-0.2%2C0.8%2C+grid%280%29%2C%0D%0Ared%28line%28-5%2C-1%2C-5%2C1%29%29%2Cred%28line%281%2C-1%2C1%2C1%29%29%2C%0D%0Agreen%28line%2819%2C-1%2C19%2C1%29%29%2Clocate%28-4%2C0.6%2C%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29%3C0%29%2C%0D%0Alocate%28-12%2C0.6%2C%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29%3E0%29%2Clocate%286%2C0.6%2C%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29%3E0%29%2C%0D%0Alocate%2820%2C0.6%2C%2819-x%29%2F%28%28x%2B5%29%28x-1%29%29%3C0%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, there are two intervals where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F%28x-1%29-4%2F%28x%2B5%29%3E=0 ALIGN=MIDDLE  ALT=\"3%2F%28x-1%29-4%2F%28x%2B5%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F%28x-1%29%3E=4%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"3%2F%28x-1%29%3E=4%2F%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-5 ALIGN=MIDDLE  ALT=\"x%3C-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%3Cx%3C=19 ALIGN=MIDDLE  ALT=\"-1%3Cx%3C=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In interval notation, we would write that as<BR>\n" );
document.write( "(&#8722;&#8734;, &#8722;5) &#8746; (1, 19] .\r<BR>\n" );
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document.write( "VERY IMPORTANT NOTE:<BR>\n" );
document.write( "Beware of \"invisible parentheses\". They have tripped many people before.<BR>\n" );
document.write( "I once saw an experienced scientist, with an advanced degree, overlook one set of those \"invisible parentheses\" in hasty calculations, and it caused untold trouble and embarrassment.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F%28x-1%29%3E=4%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"3%2F%28x-1%29%3E=4%2F%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > should be written as \"3/(x-1)>=4/(x+5)\" when you cannot write numerator above denominator with a long horizontal fraction bar in between.<BR>\n" );
document.write( "That long horizontal fraction implies that you first separately calculate what is above it and what is below it, before dividing.<BR>\n" );
document.write( "(2+3)/(3+2)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2B3%29%2F%283%2B2%29=5%2F5=1 ALIGN=MIDDLE  ALT=\"%282%2B3%29%2F%283%2B2%29=5%2F5=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > but 2+3/3+2=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2B3%2F3%2B2=2%2B1%2B2=5 ALIGN=MIDDLE  ALT=\"2%2B3%2F3%2B2=2%2B1%2B2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Most calculators \"know\" the rules for order of operations,<BR>\n" );
document.write( "and they will calculate  2 + 3 / 3 + 2 = 5 , if you enter it like that. </SPAN>\n" );
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