document.write( "Question 685031: Solve: |x/x-4| > (or equal to) 1. \n" ); document.write( "
Algebra.Com's Answer #424337 by Edwin McCravy(20065) $\"\"$ $\"About$
You can put this solution on YOUR website!
|x/x-4| > (or equal to) 1. \r
\n" ); document.write( "\n" ); document.write( " $\"abs%28x%2F%28x-4%29%29\"$ $\"%22%22%3E=%22%22\"$ $\"1\"$
\n" ); document.write( "
\r\n" );
document.write( "We remove absolute value bars:\r\n" );
document.write( "\r\n" );
document.write( " OR , }\r\n" );
document.write( "\r\n" );
document.write( "Solve the first inequality:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Get 0 on the right:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Critical numbers are when numerator or denominator = 0\r\n" );
document.write( "2x-4=0 gives critical number 2\r\n" );
document.write( "x-4=0 gives critical number 4\r\n" );
document.write( "\r\n" );
document.write( "----------o-----o------\r\n" );
document.write( "-1  0  1  2  3  4  5  6\r\n" );
document.write( "\r\n" );
document.write( "We test a value of x to the left of 2.  We pick 0 and\r\n" );
document.write( "substitute it into the original inequality\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "That is false so we do not shade the number line to the\r\n" );
document.write( "left of 2.\r\n" );
document.write( "\r\n" );
document.write( "We test a value of x between 2 and 4.  We pick 3 and\r\n" );
document.write( "substitute it into the original inequality\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "That is true so we shade the number line between 2 and 4.\r\n" );
document.write( "\r\n" );
document.write( "----------o=====o------\r\n" );
document.write( "-1  0  1  2  3  4  5  6\r\n" );
document.write( "\r\n" );
document.write( "We test a value of x to the right of 4.  We pick 5 and\r\n" );
document.write( "substitute it into the original inequality\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "That is true so we shade the number line to the\r\n" );
document.write( "right of 4.\r\n" );
document.write( "\r\n" );
document.write( "----------o=====o=====>\r\n" );
document.write( "-1  0  1  2  3  4  5  6\r\n" );
document.write( "\r\n" );
document.write( "Now we text the critical numbers in the original inequality\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This is true so we darken the circle at the value 2\r\n" );
document.write( "\r\n" );
document.write( "----------@=====o=====>\r\n" );
document.write( "-1  0  1  2  3  4  5  6\r\n" );
document.write( "\r\n" );
document.write( "We do not darken the circle at 4 because the denominator in the\r\n" );
document.write( "original inequality cannot be zero.\r\n" );
document.write( "\r\n" );
document.write( "Now we consider the other inequality\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Get 0 on the right:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The only critical number this time is \r\n" );
document.write( "x-4=0 gives critical number 4.\r\n" );
document.write( "\r\n" );
document.write( "We have already considered that critical number\r\n" );
document.write( "in the first part so nothing is gained from this \r\n" );
document.write( "second inequality.  Therefore the solution is \r\n" );
document.write( "\r\n" );
document.write( "----------@=====o=====>\r\n" );
document.write( "-1  0  1  2  3  4  5  6\r\n" );
document.write( "\r\n" );
document.write( "Which has interval notation\r\n" );
document.write( "\r\n" );
document.write( "         [2, 4) U (4, )\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );