document.write( "Question 1144798: Email - aurora.0723@outlook.com
\n" ); document.write( "Question - when trying to determine the x value of an inequality, why do we sometime flip the inequality even though it is not being divided the zero. For example,
\n" ); document.write( " Solve [1/(x-1)] < 5
\n" ); document.write( "Case 1 = {x-1<0 x<1
\n" ); document.write( " { (1/x-1)<5 (1/x-1)>5 x<(6/5)
\n" ); document.write( "Then you do the second case. However, in the first case, why does (1/x-1)<5 turn to (1/x-1)>5? Why does the inequality flip?
\n" ); document.write( "Thanks for the help \n" ); document.write( "
Algebra.Com's Answer #765978 by greenestamps(13334)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "\"1%2F%28x-1%29%3C5\"

\n" ); document.write( "Solution strategy: multiply both sides by (x-1) to clear fractions.

\n" ); document.write( "Case 1: x-1 < 0 (i.e., x < 1)

\n" ); document.write( "If x < 1, (x-1) is negative, so multiplying by (x-1) reverses the direction of the inequality.

\n" ); document.write( "\"1+%3E+5x-5\"
\n" ); document.write( "\"6+%3E+5x\"
\n" ); document.write( "\"5x+%3C+6\"
\n" ); document.write( "\"x+%3C+6%2F5\"

\n" ); document.write( "So if x < 1, an \"additional\" restriction is that x < 6/5. But ALL values of x that are less than 1 are less than 6/5. So all numbers less than 1 are part of the solution set.

\n" ); document.write( "Case 2: x-1 > 0 (i.e., x > 1)

\n" ); document.write( "If x > 1, (x-1) is positive, so multiplying by (x-1) does not change the direction of the inequality.

\n" ); document.write( "\"1+%3C+5x-5\"
\n" ); document.write( "\"6+%3C+5x\"
\n" ); document.write( "\"6%2F5+%3C+x\"
\n" ); document.write( "\"x+%3E+6%2F5\"

\n" ); document.write( "So if x > 1, the additional restriction is that x > 6/5. So the other part of the solution set is all the number greater than 6/5 = 1.2.

\n" ); document.write( "Final solution set: (-infinity, 1) union (1.2, infinity) \n" ); document.write( "
\n" );