document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=180067>Question 180067</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0321233832/algebrahomeworkh>elementry and intermediate algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> 1.	Explain the following in your own words: (a + b)2 &#8800; a2 + b2. Give a numerical example to illustrate this. \r<BR>\n" );
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document.write( "3.	Add -5 to both sides of the inequality 7 > 1. Then divide both sides of the inequality -25 > -30 by -6. Explain in your own words what happens in both examples and when the direction of the inequality symbol is reversed. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #134949 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=134949\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "I can explain these things in my words, but if you are going to use this for your homework, you'll have to change them and still manage to communicate the same ideas in order for you to properly answer the questions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (a + b)^2 \neq a^2 + b^2\">\r<BR>\n" );
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document.write( "Obviously, these two expressions <i><b>are</b></i> equal if either <i><b>a</b></i>, <i><b>b</b></i>, or both are 0.  The question is, do there exist any non-zero values for <i><b>a</b></i> and <i><b>b</b></i> that exhibit the property <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (a + b)^2 = a^2 + b^2\">? \r<BR>\n" );
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document.write( "Let's assume that there exists a number <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a \neq 0\"> and a number <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b \neq 0\"> such that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (a + b)^2 = a^2 + b^2\">.\r<BR>\n" );
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document.write( "We can add <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -b^2\"> to both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{          }(a + b)^2 - b^2 = a^2\">.\r<BR>\n" );
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document.write( "Then we can factor the difference of two squares on the left:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{          } ((a + b) - b)((a + b) + b) = a^2\">.\r<BR>\n" );
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document.write( "Collect terms and distribute:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{          } a(a + 2b) = a^2\">.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{          } a^2 + 2ab = a^2\">.\r<BR>\n" );
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document.write( "Now add <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -a^2\"> to both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{          } 2ab = 0\">.\r<BR>\n" );
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document.write( "According to the Zero Product Rule, if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2ab = 0\"> then either <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a = 0\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b = 0\">, but that contradicts our original assumption that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a \neq 0\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b \neq 0\">.  Hence, <i>reductio ad absurdum</i> there are no non-zero values for <i>a</i> and <i>b</i> such that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (a + b)^2 = a^2 + b^2\">.  Therefore, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (a + b)^2 \neq a^2 + b^2 \text{       } \math \forall a \neq 0 \text{ and } \math b \neq 0\"> \r<BR>\n" );
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document.write( "Having said all of that, finding a numerical example to illustrate <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (a + b)^2 \neq a^2 + b^2\"> should be a fairly simple task.  I'll leave that part to you.\r<BR>\n" );
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document.write( "Add -5 to the inequality 7 > 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -5 + 7 > -5 + 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2 > -4\"> which is a true statement without changing the sense of the inequality.\r<BR>\n" );
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document.write( "Then divide both sides of the inequality -25 > -30 by -6.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac {-25}{-6} > \frac {-30}{-6}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac {25}{6} > \frac {30}{6}\">\r<BR>\n" );
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document.write( "Which is now a false statement.  Since we divided by a number less than zero, the sense of the inequality must be reversed.  This is also true when you multiply by a negative, since dividing by a number is the same as multiplying by the reciprocal.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac {25}{6} < \frac {30}{6}\">\r<BR>\n" );
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document.write( "Having reversed the sense of the inequality, the statement is again true.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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