document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=384383>Question 384383</A>: <I><SPAN STYLE=\"word-break: break-all;\"> B is taller than j and 3 pillars. P is shorter than B and 2 pillars is j shorter/taller than P? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #272128 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=272128\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let > imply taller<BR>\n" );
document.write( "Let < imply shorter\r<BR>\n" );
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document.write( "the first statement says that:\r<BR>\n" );
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document.write( "B > j + 3\r<BR>\n" );
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document.write( "This means that there is a positive value we can add to j + 3 to make it equal to B.\r<BR>\n" );
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document.write( "We'll call that value x.\r<BR>\n" );
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document.write( "We get:\r<BR>\n" );
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document.write( "B = j + 3 + x\r<BR>\n" );
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document.write( "The second statement says that:\r<BR>\n" );
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document.write( "P < B + 2\r<BR>\n" );
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document.write( "We can replace B with j + 3 + x to make that statement become:\r<BR>\n" );
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document.write( "P < j + 3 + x + 2\r<BR>\n" );
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document.write( "We can simplify this statement to be equal to:\r<BR>\n" );
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document.write( "P < j + x + 5\r<BR>\n" );
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document.write( "This means that there is a value we can add to P to make it equal to j + x + 5.\r<BR>\n" );
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document.write( "We'll call that value y.\r<BR>\n" );
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document.write( "The statement becomes:\r<BR>\n" );
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document.write( "P + y = j + x + 5\r<BR>\n" );
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document.write( "If y = x + 5, then the statement will become:\r<BR>\n" );
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document.write( "P + x + 5 = j + x + 5 \r<BR>\n" );
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document.write( "If we subtract (x + 5) from both sides of the equation, we will then get:\r<BR>\n" );
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document.write( "P = j\r<BR>\n" );
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document.write( "So, if y = x + 5, then we get P = j.\r<BR>\n" );
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document.write( "So P = j is possible when y = x + 5.\r<BR>\n" );
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document.write( "Going back to the original equation, we have:\r<BR>\n" );
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document.write( "P + y = j + x + 5\r<BR>\n" );
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document.write( "If we let y be greater than x + 5, then we get another value (call it z), such that:\r<BR>\n" );
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document.write( "y = x + 5 + z\r<BR>\n" );
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document.write( "Our equation of P + y = j + x + 5 becomes:\r<BR>\n" );
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document.write( "P + x + 5 + z = j + x + 5\r<BR>\n" );
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document.write( "If we subtract (x + 5) from both sides of this equation, we get:\r<BR>\n" );
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document.write( "P + z = j\r<BR>\n" );
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document.write( "This implies that P < j\r<BR>\n" );
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document.write( "So we can have P = j and we can have P < j\r<BR>\n" );
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document.write( "Going back to the original equation again, we have:\r<BR>\n" );
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document.write( "P + y = j + x + 5\r<BR>\n" );
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document.write( "If we let y be smaller than x + 5, then we get another value (call it w), such that:\r<BR>\n" );
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document.write( "y + w = x + 5 which becomes y = x + 5 - w after we subtract w from both sides of the equation.\r<BR>\n" );
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document.write( "Our equation becomes:\r<BR>\n" );
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document.write( "P + x + 5 - w = j + x + 5\r<BR>\n" );
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document.write( "If we subtract x + 5 from both sides of this equation, we get:\r<BR>\n" );
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document.write( "P - w = j\r<BR>\n" );
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document.write( "If we add w to both sides of this equation, we get:\r<BR>\n" );
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document.write( "P = j + w\r<BR>\n" );
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document.write( "This implies that P > j\r<BR>\n" );
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document.write( "So, we can get all 3 conditions, depending on the relationship between x and y.\r<BR>\n" );
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document.write( "They are:\r<BR>\n" );
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document.write( "P = j when y = x + 5<BR>\n" );
document.write( "P < j when y > x + 5<BR>\n" );
document.write( "P > j when y < x + 5\r<BR>\n" );
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document.write( "We can put this into numbers in order to confirm that what we have determined algebraically is correct.\r<BR>\n" );
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document.write( "Our starting statements are:\r<BR>\n" );
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document.write( "B > j + 3<BR>\n" );
document.write( "P < B + 2\r<BR>\n" );
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document.write( "We translated these into:\r<BR>\n" );
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document.write( "B = j + 3 + x (x > 0)<BR>\n" );
document.write( "P + y = B + 2 (y > 0)\r<BR>\n" );
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document.write( "Let's let x = 5 and y = 10 (y = x + 5)\r<BR>\n" );
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document.write( "We get:\r<BR>\n" );
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document.write( "B = j + 3 + 5 which becomes B = j + 8<BR>\n" );
document.write( "P + 10 = B + 2 which becomes P = B - 8\r<BR>\n" );
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document.write( "Substitute j + 8 for B and we get P = j + 8 - 8 which becomes P = j.\r<BR>\n" );
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document.write( "When y = x + 5, we get P = j.\r<BR>\n" );
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document.write( "Back to the original equations:\r<BR>\n" );
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document.write( "B > j + 3<BR>\n" );
document.write( "P < B + 2\r<BR>\n" );
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document.write( "We translated these into:\r<BR>\n" );
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document.write( "B = j + 3 + x (x > 0)<BR>\n" );
document.write( "P + y = B + 2 (y > 0)\r<BR>\n" );
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document.write( "Let's let x = 5 and y = 11 (y > x + 5)\r<BR>\n" );
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document.write( "We get:\r<BR>\n" );
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document.write( "B = j + 3 + 5 which becomes B = j + 8<BR>\n" );
document.write( "P + 11 = B + 2\r<BR>\n" );
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document.write( "Substitute j + 8 for B in the second equation and we get:\r<BR>\n" );
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document.write( "P + 11 = j + 8 + 2 which becomes:<BR>\n" );
document.write( "P + 11 = j + 10<BR>\n" );
document.write( "Subtract 10 from both sides of the equation to get:<BR>\n" );
document.write( "P + 1 = j<BR>\n" );
document.write( "This implies that P is less than j (P < j).\r<BR>\n" );
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document.write( "Back to the original equations:\r<BR>\n" );
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document.write( "B > j + 3<BR>\n" );
document.write( "P < B + 2\r<BR>\n" );
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document.write( "We translated these into:\r<BR>\n" );
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document.write( "B = j + 3 + x (x > 0)<BR>\n" );
document.write( "P + y = B + 2 (y > 0)\r<BR>\n" );
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document.write( "Let's let x = 5 and y = 9 (y < x + 5)\r<BR>\n" );
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document.write( "We get:\r<BR>\n" );
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document.write( "B = j + 3 + 5 which becomes B = j + 8<BR>\n" );
document.write( "P + 9 = B + 2\r<BR>\n" );
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document.write( "Substitute for B in the second equation to get:\r<BR>\n" );
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document.write( "P + 9 = j + 8 + 2 which becomes:<BR>\n" );
document.write( "P + 9 = j + 10\r<BR>\n" );
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document.write( "Subtract 9 from both sides of the equation to get:\r<BR>\n" );
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document.write( "P = j + 1\r<BR>\n" );
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document.write( "This implies that P > j\r<BR>\n" );
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document.write( "Bottom line:\r<BR>\n" );
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document.write( "P can be taller than or equal to or shorter than j.\r<BR>\n" );
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document.write( "This all depends on the relationship between x and y.\r<BR>\n" );
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document.write( "In the translation of the equation B > j + 3 to B = j + 3 + x, x is the amount that B is greater than (taller than) j + 3.\r<BR>\n" );
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document.write( "In the translation of the equation P < B + 2, to P + y = B + 2, y is the amount that P is smaller than (shorter than) B + 2.\r<BR>\n" );
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document.write( "Making the inequality an equality allows the problem to be solved.\r<BR>\n" );
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document.write( "The creation of the difference variables of x, y, z, w allowed the equality to be modeled from the inequality.\r<BR>\n" );
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document.write( "x,y,z,w are all assumed to be greater than 0.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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