document.write( "Question 82826: I need help with this equation:
\n" ); document.write( "(x-5)exp-5<_0 does the negative 5 exponent tell me that there may be no solution? \n" ); document.write( "
Algebra.Com's Answer #59404 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I need help with this equation:\r\n" );
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document.write( "It's not an equation, but an inequality. \r\n" );
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document.write( "(x-5)exp-5<_0 does the negative 5 exponent tell me that there may be no solution?\r\n" );
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document.write( "No, not at all!\r\n" );
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document.write( "\"%28x-5%29%5E%28-5%29\" < 0\r\n" );
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document.write( "just means:\r\n" );
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document.write( "\"1%2F%28x-5%29%5E5\" < 0, undefined for x = 5\r\n" );
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document.write( "Now we take the fifth root of\r\n" );
document.write( "both sides.  Remember there is only\r\n" );
document.write( "one real odd root of any real number,\r\n" );
document.write( "regardless of whether the number under\r\n" );
document.write( "the radical is is positive, negative, \r\n" );
document.write( "or zero.  The same can't be said for \r\n" );
document.write( "even roots, but we don't have to worry\r\n" );
document.write( "about them here since 5th root is an \r\n" );
document.write( "odd root.\r\n" );
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document.write( "\"root%285%2C+1%2F%28x-5%29%5E5%29\" < \"root%285%2C0%29\", undefined for x = 5\r\n" );
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document.write( "\"1%2F%28x-5%29\"  < \"0\", undefined for x = 5\r\n" );
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document.write( "There is no value of x for which \"1%2F%28x-5%29\" = \"0\", therefore\r\n" );
document.write( "we may as well write: \r\n" );
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document.write( "\"1%2F%28x-5%29\" < \"0\",  undefined for x = 5\r\n" );
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document.write( "Then we know that a number is positive if and only if its reciprocal\r\n" );
document.write( "is positive, and negative if and only if its reciprocal is negative.    \r\n" );
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document.write( "So x - 5 < 0 or \r\n" );
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document.write( "     x < 5\r\n" );
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document.write( "The graph of that looks like this:\r\n" );
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document.write( "<=========================o------------------>\r\n" );
document.write( " -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 \r\n" );
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document.write( "The solution is set-builder notation is\r\n" );
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document.write( "{x|x < 5}\r\n" );
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document.write( "Or in interval notation:  (-oo, 5)\r\n" );
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document.write( "The graph of the left side of the original \r\n" );
document.write( "inequality\r\n" );
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document.write( "y = \"%28x-5%29%5E%28-5%29\"\r\n" );
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document.write( "is this \"graph%28400%2C400%2C+-10%2C10%2C-10%2C10%2C1%2F%28x-5%29%5E5%2C9999%28x-5%29%29\"\r\n" );
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document.write( "The curve is always below the x-axis when x is less than 5\r\n" );
document.write( "and always above the x-axis when x is greater than 5.  I agree\r\n" );
document.write( "it looks as if it touches the x-axis, but it doesn't really.\r\n" );
document.write( "It just comes so close to it that it can only be drawn looking \r\n" );
document.write( "as though it touches it but it doesn't.  There is a vertical \r\n" );
document.write( "asymptote at x = 5 as you see.  There is no value of x\r\n" );
document.write( "that will produce equality, hence this convinces us that the\r\n" );
document.write( "interval notation\r\n" );
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document.write( "(-oo, 5) is the correct solution.\r\n" );
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document.write( "Edwin
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