document.write( "Question 647930: |x-3| is less than or equal to k. Find a value of k for which the inequality has no solution. Find a value for which the inequality has exactly one solution. Find a value of k for which a solution exists but for which the solution set does not include 5. \n" ); document.write( "
Algebra.Com's Answer #406487 by Edwin McCravy(20056)\"\" \"About 
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|x-3| ≦ k. Find a value of k for which the inequality has no solution.
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document.write( "An absolute value can NEVER equal to a negative number, so this\r\n" );
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document.write( "|x-3| ≦ -1\r\n" );
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document.write( "has no solution.  (Any negative number on the right will do)\r\n" );
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document.write( "Answer: when k = -1 or any other negative number.\r\n" );
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Find a value for which the inequality has exactly one solution.
\n" ); document.write( "Find a value of k for which a solution exists but for which the solution set does not include 5.
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document.write( "Both those problems involve cases where a solution exists, which will be\r\n" );
document.write( "when k is a non-negative number. \r\n" );
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document.write( "Let k be a non-negative number, that is, k ≧ 0, so that the\r\n" );
document.write( "inequality will have a solution.\r\n" );
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document.write( "Then |x-3| ≦ k will become this three sided inequality:\r\n" );
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document.write( "      -k ≦ x-3 ≦ k  where   x ≧ 0\r\n" );
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document.write( "Add 3 to all 3 sides:\r\n" );
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document.write( "     3-k ≦ x ≦ k+3\r\n" );
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document.write( "    The solution is the closed interval [3-k, k+3]\r\n" );
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document.write( "Find a value for which the inequality has exactly one solution.\r\n" );
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document.write( "The closed interval [3-k, k+3] will have infinitely many solutions\r\n" );
document.write( "except when the interval shrinks to just one point, and that will be \r\n" );
document.write( "when its endpoints are equal, so we set them equal:\r\n" );
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document.write( "     3-k = k+3\r\n" );
document.write( "     -2k = 0\r\n" );
document.write( "       k = 0\r\n" );
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document.write( "So |x-3| ≦ 0 has exactly one solution, when k=0\r\n" );
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document.write( "    3-0 ≦ x ≦ 0+3\r\n" );
document.write( "      3 ≦ x ≦ 3\r\n" );
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document.write( "Which means that solution is x = 3.\r\n" );
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document.write( "Answer: when k=0, there is exactly one solution, x = 3.\r\n" );
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document.write( "Find a value of k for which a solution exists but for which the solution set does not include 5.\r\n" );
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document.write( "    The solution is the closed interval [3-k, k+3]\r\n" );
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document.write( "It will not contain k if k is either \r\n" );
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document.write( "(a) less than the left endpoint\r\n" );
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document.write( "      k < 3-k\r\n" );
document.write( "     2k < 3\r\n" );
document.write( "      k < \"3%2F2\"\r\n" );
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document.write( "That is 0 ≦ k < \"3%2F2\"\r\n" );
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document.write( "So we can pick any value for k in the interval  [0, \"3%2F2\"), say k=1\r\n" );
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document.write( "[actually the value k=0 that we used above would be a possible answer\r\n" );
document.write( "here because the one-point interval [3-0,0+3] or [3,3} does not contain \r\n" );
document.write( "k=0.]\r\n" );
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document.write( "or\r\n" );
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document.write( "(b) greater than the right endpoint\r\n" );
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document.write( "    k > k+3 \r\n" );
document.write( "    0 > 3\r\n" );
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document.write( "That is a contradiction so we discard this case.\r\n" );
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document.write( "So k = 1 is a good answer for this.\r\n" );
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document.write( "Edwin
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