document.write( "Question 880568: If a^3b^5c^7 < 0 and b > 0, which of the following must be true?\r
\n" ); document.write( "\n" ); document.write( "I. ab < 0
\n" ); document.write( "II. ac < 0
\n" ); document.write( "III. a^2c^2 > 0
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #531564 by KMST(5328) $\"\"$ $\"About$

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Since $\"a%5E3b%5E5c%5E7%3C0\"$ , we know that $\"a%5E3b%5E5c%5E7%3C%3E0\"$ , so $\"system%28a%3C%3E0%2Cb%3C%3E0%2C%22and%22%2Cc%3C%3E0%29\"$ .
\n" ); document.write( "That is good to know, in case we want to divide by those numbers.
\n" ); document.write( "For all $\"a%3C%3E0\"$ and $\"c%3C%3E0\"$ , $\"a%5E2%3E0\"$ and $\"c%5E6=%28c%5E3%29%5E2%3E0\"$ , because they are squares.
\n" ); document.write( "If $\"b%3E0\"$ , then $\"b%5E5%3E0\"$ .
\n" ); document.write( "So $\"a%5E2\"$ , $\"b%5E5\"$ and $\"c%5E6\"$ are positive, and so is their product, $\"a%5E2b%5E5c%5E6\"$ .
\n" ); document.write( "Then, starting with
\n" ); document.write( " $\"a%5E3b%5E5c%5E7%3C0\"$ ,
\n" ); document.write( "we can divide both sides of the inequality by that positive product:
\n" ); document.write( " $\"a%5E3b%5E5c%5E7%2Fa%5E2b%5E5c%5E6\"$ $\"%22%3C%22\"$ $\"0%2Fa%5E2b%5E5c%5E6\"$
\n" ); document.write( "That simplifies to $\"highlight%28ac%3C0%29\"$
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\n" ); document.write( "NOTE:
\n" ); document.write( "Of course, $\"a%5E2c%5E2%3E0\"$ also must be true if $\"system%28a%3C%3E0%2C%22and%22%2Cc%3C%3E0%29\"$ , which is a consequence of $\"a%5E3b%5E5c%5E7%3C0\"$ .
\n" ); document.write( "So for $\"a%5E3b%5E5c%5E7%3C0\"$ , $\"a%5E2c%5E2%3E0\"$ is true if $\"b%3E0\"$ , and it is also true if $\"b%3C0\"$ ,
\n" ); document.write( "but III is not the answer they want,
\n" ); document.write( "and the answer that will help your grade is $\"highlight%28II%29\"$ . \n" ); document.write( "

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