document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200141>Question 1200141</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following inequalities : <BR>\n" );
document.write( "a) x^2 - 5x + 4 > 0<BR>\n" );
document.write( "b) k(2k-3) < k^2 - 2<BR>\n" );
document.write( "c) x^2 - 5x - 14 > 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834177 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=834177\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll do the first problem to get you started.\r<BR>\n" );
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document.write( "Let's solve the associated equation<BR>\n" );
document.write( "x^2 - 5x + 4 = 0\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "x^2 - 5x + 4 = 0<BR>\n" );
document.write( "(x-1)(x-4) = 0<BR>\n" );
document.write( "x-1 = 0 or x-4 = 0<BR>\n" );
document.write( "x = 1 or x = 4<BR>\n" );
document.write( "The quadratic formula is an alternative pathway.\r<BR>\n" );
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document.write( "The roots of x^2 - 5x + 4 are x = 1 and x = 4\r<BR>\n" );
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document.write( "Draw out a number line to plot 1 and 4 on it.\r<BR>\n" );
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document.write( "Label the following regions<BR>\n" );
document.write( "A: stuff to the left of 1<BR>\n" );
document.write( "B: stuff between 1 and 4<BR>\n" );
document.write( "C: stuff to the right of 4\r<BR>\n" );
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document.write( "Pick a value from region A.<BR>\n" );
document.write( "I'll select x = 0<BR>\n" );
document.write( "Plug it into x^2 - 5x + 4 and simplify to get...<BR>\n" );
document.write( "x^2 - 5x + 4 <BR>\n" );
document.write( "0^2 - 5*0 + 4 <BR>\n" );
document.write( "4<BR>\n" );
document.write( "We arrive at a positive value.<BR>\n" );
document.write( "Therefore, x = 0 makes x^2 - 5x + 4 > 0 to be true<BR>\n" );
document.write( "Any other value in region A will arrive at this same conclusion.<BR>\n" );
document.write( "Region A is part of the solution set.\r<BR>\n" );
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document.write( "Now onto region B.<BR>\n" );
document.write( "Let's pick x = 2 as a representative value from this region 1 < x < 4<BR>\n" );
document.write( "x^2 - 5x + 4<BR>\n" );
document.write( "2^2 - 5*2 + 4<BR>\n" );
document.write( "4 - 10 + 4<BR>\n" );
document.write( "-2<BR>\n" );
document.write( "The result is negative, which means x^2 - 5x + 4 > 0 would be false for any x value in region B.<BR>\n" );
document.write( "Region B is NOT part of the solution set.\r<BR>\n" );
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document.write( "Lastly region C, which is x > 4<BR>\n" );
document.write( "Let's pick x = 5<BR>\n" );
document.write( "x^2 - 5x + 4<BR>\n" );
document.write( "5^2 - 5*5 + 4<BR>\n" );
document.write( "25 - 25 + 4<BR>\n" );
document.write( "4<BR>\n" );
document.write( "Like with region A, we get a positive result to show x^2 - 5x + 4 > 0 would be true for x values in this interval.<BR>\n" );
document.write( "Region C is part of the solution set.\r<BR>\n" );
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document.write( "We found that<BR>\n" );
document.write( "Region A, x < 1, is part of the solution set<BR>\n" );
document.write( "Region C, x > 4, is also part of the solution set\r<BR>\n" );
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document.write( "Collectively the entire solution set is x < 1 or x > 4<BR>\n" );
document.write( "We can write this in interval notation to get (-infinity, 1) U (4 < infinity)<BR>\n" );
document.write( "The U symbol is the union operator. It glues together the two intervals in an \"or\" fashion.<BR>\n" );
document.write( "We are either in the interval (-infinity, 1) OR we are in the interval (4 < infinity). We cannot be in both intervals at the same time.\r<BR>\n" );
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document.write( "Answer as inequalities: <font color=red>x < 1 or x > 4</font><BR>\n" );
document.write( "Answer as interval notation: <font color=red>(-infinity, 1) U (4 < infinity)</font>\r<BR>\n" );
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document.write( "Similar question is found here<BR>\n" );
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