document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=629095>Question 629095</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (x+5)(x+6)(x+2)>0<BR>\n" );
document.write( "Select the correct choice below, and if necessary, fill in the answer box to complete your choice.\r<BR>\n" );
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document.write( "A.  The solution set is {x/________}<BR>\n" );
document.write( "(Use at least one inequality or compound inequality to express your answer.  For answers with more one inequality, separate the inequalities by a comma or the word  \"or\".\r<BR>\n" );
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document.write( "B.  The solutions are all real numbers\r<BR>\n" );
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document.write( "C.  There is no solution. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #396040 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=396040\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (x+5)(x+6)(x+2)>0<BR>\n" );
document.write( "This inequality, in essence, says that the product of three numbers is greater than zero. A synonym for \"greater than zero\" is \"positive\". So we have an inequality that says that the product of three numbers is positive.<br><BR>\n" );
document.write( "Think about that. Three numbers multiplied with a positive result. How/when does this happen? With a little thought I hope you can realize that none of the three numbers could be zero because then the product would be zero, not positive. And with a little more thought I hope it is easy to see that there only two ways a product of three numbers is positive:<ul><li>All three factors are positive; or</li><li>Two factors are negative and the third one is positive.</li></ul>Now we just have to translate this logic:<BR>\n" );
document.write( "(All three factors are positive) or (Two factors are negative and the third one is positive)<BR>\n" );
document.write( "into mathematical expressions we can use to solve the problem. Translating<BR>\n" );
document.write( "\"All three factors are positive\"<BR>\n" );
document.write( "we get:<BR>\n" );
document.write( "(x+5 > 0 and x+6 > 0 and x+2 > 0)<br><BR>\n" );
document.write( "Translating<BR>\n" );
document.write( "\"Two factors are negative and the third one is positive\"<BR>\n" );
document.write( "is a little more challenging. The straightforward way is to express every possible combination:<BR>\n" );
document.write( "((x+5 < 0 and x+6 < 0 and x+2 > 0) or (x+5 < 0 and x+6 > 0 and x+2 < 0) or (x+5 > 0 and x+6 < 0 and x+2 < 0))<br><BR>\n" );
document.write( "So all together<BR>\n" );
document.write( "(All three factors are positive) or (Two factors are negative and the third one is positive)<BR>\n" );
document.write( "translates into<BR>\n" );
document.write( "(x+5 > 0 and x+6 > 0 and x+2 > 0) or ((x+5 < 0 and x+6 < 0 and x+2 > 0) or (x+5 < 0 and x+6 > 0 and x+2 < 0) or (x+5 > 0 and x+6 < 0 and x+2 < 0))<br><BR>\n" );
document.write( "We could go ahead and solve this. But let's try to use our brains a little bit to make this much easier. If at any point below you don't understand what I'm doing then just go back to the big mess above and solve it.<br><BR>\n" );
document.write( "Let's think about the three factors:<BR>\n" );
document.write( "x+5, x+6 and x+2<BR>\n" );
document.write( "I hope that it is easy to see that no matter what number x might be,<BR>\n" );
document.write( "x+6 will be the largest factor,<BR>\n" );
document.write( "x+5 will be the next largest factor and<BR>\n" );
document.write( "x+2 will be the smallest factor<br><BR>\n" );
document.write( "We can use this to our advantage. For example, we want to express \"All three factors are positive.\" I hope that, after a little thought, it will be clear that if the smallest of the factors is positive the other two would have to be positive, too. So to express \"All three factors are positive\" all we really need is:<BR>\n" );
document.write( "(x+2 > 0)<br><BR>\n" );
document.write( "More importantly we can use this logic to simplify our expression for \"Two factors are negative and the third one is positive\". If only one factor is to be positive, doesn't it need to be the largest one (x+6)? And if the other two have to be negative, isn't it enough that the larger of the two (x+5) be negative? So for \"Two factors are negative and the third one is positive\" all we really need is:<BR>\n" );
document.write( "x+6 > 0 and x+5 < 0<br><BR>\n" );
document.write( "So with this additional logic<BR>\n" );
document.write( "(All three factors are positive) or (Two factors are negative and the third one is positive)<BR>\n" );
document.write( "can be translated into just:<BR>\n" );
document.write( "(x+2 > 0) or (x+6 > 0 and x+5 < 0)<BR>\n" );
document.write( "instead of<BR>\n" );
document.write( "(x+5 > 0 and x+6 > 0 and x+2 > 0) or ((x+5 < 0 and x+6 < 0 and x+2 > 0) or (x+5 < 0 and x+6 > 0 and x+2 < 0) or (x+5 > 0 and x+6 < 0 and x+2 < 0))<br><BR>\n" );
document.write( "Both of the above work and, believe it or not, give the same solution. But I hope you agree that the extra thinking is worth how simple the \"smart\" version will be to solve. I'm only going to finish the simple version:<BR>\n" );
document.write( "(x+2 > 0) or (x+6 > 0 and x+5 < 0)<BR>\n" );
document.write( "Solving this we get:<BR>\n" );
document.write( "(x > -2) or (x > -6 and x < -5)<br><BR>\n" );
document.write( "This is clear not \"all real number\" and not \"no solution\". So the answer is \"A\". </SPAN>\n" );
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