document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196904>Question 1196904</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve each inequality, express the solution set in interval notation, then graph the solution set.\r<BR>\n" );
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document.write( "1.<BR>\n" );
document.write( "2x/x−7 ≤ 5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829978 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13215)</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=829978\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "As tutor @ikleyn points out, if you solve the inequality by multiplying both sides of the inequality by x-7, then you need to consider two separate cases -- where x-7 is positive, and where it is negative.<br><BR>\n" );
document.write( "Here is an alternative method for solving this kind of problem.  Note in this example the algebra is probably more complicated with this method than with the other; however, it is a useful method to know, and in many similar problems the work required with this method will be easier.<br><BR>\n" );
document.write( "Instead of multiplying both side of the inequality by anything, rewrite the inequality with only \"0\" on the right; then analyze the resulting rational function to find where the inequality is satisfied.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2F%28x-7%29%3C=5 ALIGN=MIDDLE  ALT=\"2x%2F%28x-7%29%3C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2F%28x-7%29-5%3C=0 ALIGN=MIDDLE  ALT=\"2x%2F%28x-7%29-5%3C=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%2F%28x-7%29-5%28x-7%29%2F%28x-7%29%3C=0 ALIGN=MIDDLE  ALT=\"2x%2F%28x-7%29-5%28x-7%29%2F%28x-7%29%3C=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-5x%2B35%29%2F%28x-7%29%3C=0 ALIGN=MIDDLE  ALT=\"%282x-5x%2B35%29%2F%28x-7%29%3C=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3x%2B35%29%2F%28x-7%29%3C=0 ALIGN=MIDDLE  ALT=\"%28-3x%2B35%29%2F%28x-7%29%3C=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The critical points of that rational function are where the numerator or denominator is zero.  The function value is 0 at x=35/3 and undefined at x=7.<br><BR>\n" );
document.write( "Using test points on the three intervals determined by the critical points, and observing whether the function is 0 or undefined at those critical points, the solution to the inequality is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C7 ALIGN=MIDDLE  ALT=\"x%3C7\"   DISABLED_style= \"max-width:90%; height:auto;\" > U <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=35%2F3 ALIGN=MIDDLE  ALT=\"x%3E=35%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph, showing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2F%28x-7%29 ALIGN=MIDDLE  ALT=\"2x%2F%28x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) being less then 5 (green) for values of x less than 7 and greater than 35/3:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C400%2C-10%2C20%2C-10%2C10%2C2x%2F%28x-7%29%2C5%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C400%2C-10%2C20%2C-10%2C10%2C2x%2F%28x-7%29%2C5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Finally, to be technically correct in a response to your question, we should note that the above discussion (and in the responses from the other tutors) is NOT for the inequality you posted.  The inequality you posted, \"2x/x−7 ≤ 5\", is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2Fx-7%3C=5 ALIGN=MIDDLE  ALT=\"2x%2Fx-7%3C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >  -->  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-7%3C=5 ALIGN=MIDDLE  ALT=\"2-7%3C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >  -->  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%3C=5 ALIGN=MIDDLE  ALT=\"-5%3C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "which is true for all values of x for which it is defined -- i.e., for all x except x=0.<br><BR>\n" );
document.write( "I bother to make this point because, if you are working on a problem like this, then your level of mathematical knowledge should be enough to know that the proper use of parentheses is important....<br>\r<BR>\n" );
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