document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=476572>Question 476572</A>: <I><SPAN STYLE=\"word-break: break-all;\"> its all about non linear inequalities. (x-5)(x-4) is greater than or equal to 0. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852405 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(53028)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852405\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "its all about non linear inequalities. (x-5)(x-4) is greater than or equal to 0.<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The methods Theo uses to teach students in his posts very often \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;are monstrously verbose, and needlessly long.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The true Math explanations are usually much shorter, much more straightforward, \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;much clearer, and more instructive.\r<BR>\n" );
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document.write( "So, they want you solve inequality  (x-5)*(x-4) >= 0.\r\n" );
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document.write( "In this problem the function is presented as the product of two linear binomials, (x-5) and (x-4).\r\n" );
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document.write( "In order for the product of these two binomials would be non-negative, the factors/binomials \r\n" );
document.write( "should be of the same sign or zero.\r\n" );
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document.write( "First factor,  (x-5) is negative or zero in the area x <= 5 and non-negative or zero in the area x >= 5.\r\n" );
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document.write( "Second factor, (x-4) is negative or zero in the area x <= 4 and non-negative or zero in the area x >= 4.\r\n" );
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document.write( "Thus we see that both factors are negative or zero in the domain  x <= 4,\r\n" );
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document.write( "             and both factors are non-negative in the domain  x >= 5.\r\n" );
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document.write( "Thus the solution set is the union of two semi-infinite intervals  x <= 4 and x >= 5.\r\n" );
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document.write( "At this point, the solution is fully complete.\r\n" );
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document.write( "<U>ANSWER</U>.  The solution is the set  x <= 4  or  x >= 5.\r\n" );
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document.write( "         i.e. the area out of the interval between the roots of the function (but the solution set includes the roots).\r\n" );
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document.write( "To make sure that the solution is correct, you may plot the function.  See this plot below.\r\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C300%2C+-2%2C+9%2C+-2%2C+6%2C%0D%0A++++++++++++++%28x-5%29%2A%28x-4%29%29%0D%0A ALIGN=MIDDLE  ALT=\"graph%28400%2C300%2C+-2%2C+9%2C+-2%2C+6%2C%0D%0A++++++++++++++%28x-5%29%2A%28x-4%29%29%0D%0A\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "               Plot y = (x-5)*(x-4)\r\n" );
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document.write( "The function is a parabola opened upward. Its branches are in the upper (non-negative) half-plane out the interval\r\n" );
document.write( "on x-axis between x-intercepts, but including the x-intercepts.\r\n" );
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document.write( "The part of the parabola between the roots is in the lover (negative) half-plane.\r\n" );
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document.write( "So, this figure confirms the logical reasoning in algebraic solution.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "By the way, the answer in the post by @Theo, &nbsp;which is given as  &nbsp;&nbsp;x < 4, &nbsp;x > 5,  <BR>\n" );
document.write( "is &nbsp;INCORRECT: &nbsp;it excludes the roots of the function.\r<BR>\n" );
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document.write( "Actually, &nbsp;&nbsp;the roots must be included.\r<BR>\n" );
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document.write( "Couple of words about evolution of online solutions in the internet.\r<BR>\n" );
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document.write( "The era of online solutions and corresponding online websites started somewhere about the year ~ 2000.\r<BR>\n" );
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document.write( "First years it was at the mostly amateur level, but starting from the year ~ 2005, it transformed into <BR>\n" );
document.write( "a commercial enterprise.\r<BR>\n" );
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document.write( "The flow of online Math problems and online solutions was huge, and practically nobody tracked <BR>\n" );
document.write( "the quality of the solutions, as well practically nobody cared about it, because the major goal <BR>\n" );
document.write( "was to capture territory.\r<BR>\n" );
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document.write( "Due to this reason, good and correct solutions were placed side by side with unprofessional posts.\r<BR>\n" );
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document.write( "This development continued till approximately the year 2024, when the Artificial Intelligence came.\r<BR>\n" );
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document.write( "The audience of users and readers grew from thousands to millions.\r<BR>\n" );
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document.write( "But the base of solved problems for Artificial Intelligence was half of professional posts <BR>\n" );
document.write( "and half of unprofessional quality. So, the tone and the quality of many solutions produced by AI, <BR>\n" );
document.write( "were often below commonly accepted standards both in Math and in Math education.\r<BR>\n" );
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document.write( "This situation was heavily criticized by the community, and for this reason, starting from August 2025, <BR>\n" );
document.write( "the situations began to improve — the tone of many solutions changed; they are becoming increasingly acceptable.\r<BR>\n" );
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document.write( "And I'm proud to be contributing to improving the quality of online solutions.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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