document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=591026>Question 591026</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve. \r<BR>\n" );
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document.write( "1. (x^2-6x)^2 - 11(x^2-6x)-80=0\r<BR>\n" );
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document.write( "2. x-1/x-3>0\r<BR>\n" );
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document.write( "3. absolute value x^2+x-15=15 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #375461 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=375461\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> problem number 1:\r<BR>\n" );
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document.write( "(x^2 - 6x)^2 - 11(x^2-6x)-80 = 0\r<BR>\n" );
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document.write( "let y = x^2 - 6x<BR>\n" );
document.write( "equation becomes:<BR>\n" );
document.write( "y^2 - 11y - 80 = 0<BR>\n" );
document.write( "this is a quadratic equation that can be solved in the normal manner that you solve quadratic equations.\r<BR>\n" );
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document.write( "80 = 1 * 80 or 2 * 40 or 4 * 20 or 5 * 16 or 10 * 20<BR>\n" );
document.write( "1 +/- 80 = 81 or 79<BR>\n" );
document.write( "2 +/- 40 = 42 or 38<BR>\n" );
document.write( "4 +/- 20 = 24 or 16<BR>\n" );
document.write( "5 +/- 16 = 21 or 11 *****<BR>\n" );
document.write( "10 +/- 20 = 30 or 10<BR>\n" );
document.write( "looks like the combination of 5 * 16 is going to work because 5 * 16 = 80 and 16 - 5 = 11<BR>\n" );
document.write( "the 80 is the c term and the 11 is the b term.<BR>\n" );
document.write( "our factors are going to be (y + 5) * (y - 16) = 0<BR>\n" );
document.write( "if you multiply these factors out you will get:<BR>\n" );
document.write( "y^2 + 5y - 16y - 80 which will be equal to y^2 - 11y - 80 which is the quadratic equation we are trying to factor after we let (x^2-6x) equal to y.<BR>\n" );
document.write( "since (y+5) * (y-16) = 0, we set each equal to 0 to solve for y.<BR>\n" );
document.write( "we get:<BR>\n" );
document.write( "y + 5 = 0 which becomes y = -5<BR>\n" );
document.write( "y - 16 = 0 which becomes y = 16<BR>\n" );
document.write( "those are our solutions for y.<BR>\n" );
document.write( "we now need to convert those solutions of y to solutions of x.<BR>\n" );
document.write( "we had originally set y = (x^2-6x), so we now substitute (x^2 - 6x) for y to get:<BR>\n" );
document.write( "x^2 - 6x = -5<BR>\n" );
document.write( "and:<BR>\n" );
document.write( "x^2 - 6x = 16<BR>\n" );
document.write( "we now need to solve for x.<BR>\n" );
document.write( "x^2 - 6x = -5 becomes x^2 - 6x + 5 = 0 after we add 5 to both sides of the equation.<BR>\n" );
document.write( "x^2 - 6x = 16 becomes x^2 - 6x - 16 = 0 after we subtract 11 from both sides of the equation.<BR>\n" );
document.write( "x^2 - 6x + 5 factors are (x-5) * (x-1)<BR>\n" );
document.write( "x^2 - 6x - 16 factors are (x-8) * (x+2)<BR>\n" );
document.write( "we set each of these factors = to 0 to get:<BR>\n" );
document.write( "x = 5<BR>\n" );
document.write( "x = 1<BR>\n" );
document.write( "x = 8<BR>\n" );
document.write( "x = -2<BR>\n" );
document.write( "those are our possible solutions.<BR>\n" );
document.write( "we need to confirm those solutions are good by substituting for x in our original equation.<BR>\n" );
document.write( "that equation is:<BR>\n" );
document.write( "(x^2 - 6x)^2 - 11(x^2-6x)-80 = 0<BR>\n" );
document.write( "i used a graphing calculator to confirm those solutions were good.<BR>\n" );
document.write( "they are.<BR>\n" );
document.write( "without a graphing calculator you need to substitute each of those values for x in turn in the equation and solve the equation to see if the result is 0.<BR>\n" );
document.write( "example:<BR>\n" );
document.write( "when x = 1, original equation of (x^2 - 6x)^2 - 11(x^2-6x)-80 = 0 becomes:<BR>\n" );
document.write( "(1^2 - 6(1))^2 - 11(1^2 - 6(1)) - 80 = 0<BR>\n" );
document.write( "this simplifies to:<BR>\n" );
document.write( "(1 - 6)^2 - 11(1 - 6) - 80 = 0 which further simplifies to:<BR>\n" );
document.write( "(-5)^2 - 11(-5) - 80 = 0 which further simplifies to:<BR>\n" );
document.write( "25 + 55 - 80 = 0 which further simplifies to:<BR>\n" );
document.write( "80 - 80 = 0 which is true, confirming the  value of 1 for x is good.<BR>\n" );
document.write( "your solutions to this equation are:<BR>\n" );
document.write( "x = -2<BR>\n" );
document.write( "x = 1<BR>\n" );
document.write( "x = 5<BR>\n" );
document.write( "x = 8\r<BR>\n" );
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document.write( "problem number 2:\r<BR>\n" );
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document.write( "(x-1)/(x-3)>0\r<BR>\n" );
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document.write( "you can try to solve this by doing the following:<BR>\n" );
document.write( "(x-1) / (x-3) > 0<BR>\n" );
document.write( "multiply both sides of the equation by (x-3) to get:<BR>\n" );
document.write( "(x-1) = 0<BR>\n" );
document.write( "solve for x to get x > 0<BR>\n" );
document.write( "that, however, leads to only a partial solution if at all.\r<BR>\n" );
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document.write( "these types of equations need to address the numerator and the denominator separately.\r<BR>\n" );
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document.write( "find out when the numerator is equal to 0 and find out when the denominator is equal to 0.<BR>\n" );
document.write( "the numerator is equal to 0 when x = 1<BR>\n" );
document.write( "the denominator is equal to 0 when x = 3\r<BR>\n" );
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document.write( "the fraction (x-1) / (x-3) will be positive when the numerator is plus and the denominator is plus or when the numerator is minus and the denominator is minus.\r<BR>\n" );
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document.write( "the numerator is positive when x > 1 and the denominator is positive when x > 3.<BR>\n" );
document.write( "since 3 > 1, then both numerator and denominator will be positive when x > 3.\r<BR>\n" );
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document.write( "the numerator is negative when x < 1 and the denominator is negative when x < 3.<BR>\n" );
document.write( "since 1 < 3, then both numerator and denominator will be negative when x < 1\r<BR>\n" );
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document.write( "put these facts together and the fraction will be positive when x < 1 and when x > 3.\r<BR>\n" );
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document.write( "you can graph this equation to visually see what's going on.<BR>\n" );
document.write( "graph the equations of y = (x-1) / (x-3) as shown below:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28600%2C600%2C-10%2C10%2C-10%2C10%2C%28x-1%29%2F%28x-3%29%2C100%28x-3%29%29 ALIGN=MIDDLE  ALT=\"graph+%28600%2C600%2C-10%2C10%2C-10%2C10%2C%28x-1%29%2F%28x-3%29%2C100%28x-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "as you can see from the graph, there is a vertical asymptote at x = 3.<BR>\n" );
document.write( "this happens because the denominator of the equation becomes 0 when x = 3.<BR>\n" );
document.write( "i drew a vertical line there to show you where that happens.<BR>\n" );
document.write( "you can also see the the lines of the equation are positive when x is smaller than 1 and when x is greater than 3.\r<BR>\n" );
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document.write( "problem number 3:\r<BR>\n" );
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document.write( "absolute value of (x^2+x-15) = 15\r<BR>\n" );
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document.write( "the definition of an absolute value says that:\r<BR>\n" );
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document.write( "if absolute value of (x) = y, then:<BR>\n" );
document.write( "x = y if x is positive and:<BR>\n" );
document.write( "-x = y if x is negative.\r<BR>\n" );
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document.write( "x represents an expression that is within the absolute value signs.<BR>\n" );
document.write( "those are vertical lines as shown below:<BR>\n" );
document.write( "absolute value of (x) is shown as |x|.\r<BR>\n" );
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document.write( "your equation is:<BR>\n" );
document.write( "absolute value of (x^2+x-15) = 15<BR>\n" );
document.write( "this can also be shown as:<BR>\n" );
document.write( "|x^2 + x - 15| = 15\r<BR>\n" );
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document.write( "the expression within the absolute value signs is x^2 + x - 15\r<BR>\n" );
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document.write( "when that expression is positive, you get:<BR>\n" );
document.write( "x^2 + x - 15 = 15<BR>\n" );
document.write( "solve as you would any normal quadratic equation.<BR>\n" );
document.write( "subtract 15 from both sides of the equation to get:<BR>\n" );
document.write( "x^2 + x - 30 = 0<BR>\n" );
document.write( "factor to get:<BR>\n" );
document.write( "(x-5) * (x+6) = 0<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 5 or x = -6\r<BR>\n" );
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document.write( "when that expression is negative, you get:<BR>\n" );
document.write( "-(x^2 + x - 15) = 15<BR>\n" );
document.write( "multiply both sides of this equation by -1 to get:<BR>\n" );
document.write( "x^2 + x - 15 = -15<BR>\n" );
document.write( "add 15 to both sides of this equation to get:<BR>\n" );
document.write( "x^2 + x = 0<BR>\n" );
document.write( "factor to get:<BR>\n" );
document.write( "x * (x+1) = 0<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 0 or x = -1\r<BR>\n" );
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document.write( "put both solutions together to get:<BR>\n" );
document.write( "x = 0 or x = -1 or x = 5 or x = -6<BR>\n" );
document.write( "these are the possible solutions to your equation.\r<BR>\n" );
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document.write( "you need to confirm these solutions are good by substituting in the original equation.<BR>\n" );
document.write( "that equation is:<BR>\n" );
document.write( "|x^2 + x - 15| = 15\r<BR>\n" );
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document.write( "when x = 0, |x^2 + x - 15| = 15 becomes:<BR>\n" );
document.write( "|0 + 0 - 15| = 15 which becomes:<BR>\n" );
document.write( "|-15| = 15 which is true by definition of absolute value.\r<BR>\n" );
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document.write( "when x = -1, |x^2 + x - 15| = 15 becomes:<BR>\n" );
document.write( "|(-1)^2 + (-1) - 15) = 15 which becomes:<BR>\n" );
document.write( "|1 - 1 - 15| = 15 which becomes:<BR>\n" );
document.write( "|-15| = 15 which is true by definition of absolute value.\r<BR>\n" );
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document.write( "when x = 5, |x^2 + x - 15| = 15 becomes:<BR>\n" );
document.write( "|5^2 + 5 - 15| = 15 which becomes:<BR>\n" );
document.write( "|25 + 5 - 15| = 15 which becomes:<BR>\n" );
document.write( "|30 - 15| = 15 which becomes:<BR>\n" );
document.write( "|15| = 15 which is true by definition of absolute value.\r<BR>\n" );
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document.write( "when x = -6, |x^2 + x - 15| = 15 becomes:<BR>\n" );
document.write( "|(-6)^2 + (-6) - 15| = 15 which becomes:<BR>\n" );
document.write( "|36 - 6 - 15| = 15 which becomes:<BR>\n" );
document.write( "|30 - 15| = 15 which becomes:<BR>\n" );
document.write( "|15| = 15 which is true by definition of absolute value.\r<BR>\n" );
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document.write( "definition of absolute value states:<BR>\n" );
document.write( "|x| = y means:<BR>\n" );
document.write( "x = y if x is positive.<BR>\n" );
document.write( "-x = y if x is negative.\r<BR>\n" );
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document.write( "in our equation of |-15| = 15, 15 is negative, so we get -(-15) = 15 which becomes 15 = 15.<BR>\n" );
document.write( "in our equation of |15| = 15, 15 is positive, so we get 15 = 15 which is self evidently true.\r<BR>\n" );
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document.write( "you can also graph the equation of |x^2 + x - 15| = 15 so see what it looks like on a graph.<BR>\n" );
document.write( "subtract 15 from both sides of that equation to get:<BR>\n" );
document.write( "|x^2 + x - 15| - 15 = 0<BR>\n" );
document.write( "replace 0 with y in that equation to get:<BR>\n" );
document.write( "y = |x^2 + x - 15| = 0 and graph that equation.<BR>\n" );
document.write( "you will get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-20%2C20%2C-20%2C20%2Cabs%28x%5E2+%2B+x+-+15%29+-+15%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-20%2C20%2C-20%2C20%2Cabs%28x%5E2+%2B+x+-+15%29+-+15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "we'll take a closer in look so you can see the points where the graph crosses the x-axis.<BR>\n" );
document.write( "those points are your solution points.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-20%2C20%2C-1%2C1%2Cabs%28x%5E2+%2B+x+-+15%29+-+15%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-20%2C20%2C-1%2C1%2Cabs%28x%5E2+%2B+x+-+15%29+-+15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you can see that the graph crosses the x-axis at x = -6, -1, 0, and 5.<BR>\n" );
document.write( "this graph is the graph of the absolute value of (x^2 + x - 15) - 15.<BR>\n" );
document.write( "this is not the same as the graph of x^2 + x - 15 - 15 = 0<BR>\n" );
document.write( "the graph of that equation looks like this:<BR>\n" );
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