Algebra.Com's Answer #718790 by greenestamps(13198)![\"\"](\"/images/stars/stars-13.gif\")  You can put this solution on YOUR website! \n" ); document.write( "The other tutor's solution is wrong.... \n" ); document.write( "The given inequality is \n" ); document.write( " \n" ); document.write( "The other tutor simply multiplied both sides of the inequality by (x-1). That won't give you the full solution set; and it might give you a solution set which is NOT right. The problem with that method is that (x-1) might be negative; if it is, the direction of the inequality changes. \n" ); document.write( "You can solve the problem by multiplying both sides of the inequality by (x-1); but you need to look at two cases -- when (x-1) is positive and when it is negative. \n" ); document.write( "Case 1: If (x-1) is positive (that is, x>1), then \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "That is the \"solution\" that the other tutor gave; but it is not a solution at all. For this case 1, we are only considering solutions with x>1; \"x <= 0\" does not give any solutions with x>1. \n" ); document.write( "So this first case gives us NO solutions to the inequality. \n" ); document.write( "Case 2: If (x-1) is negative (that is, x<1), then \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "Since case 2 only considers values of x less than 1, the solution set for this case is 0 <= x < 1. \n" ); document.write( "So the complete solution set is 0 <= x < 1; or, in interval notation, [0,1). \n" ); document.write( "As an alternative to the above method of solution where we consider two different cases, we can modify the given inequality so that one side of the inequality is 0. Following is a solution to the problem using that strategy. \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "For x values less than 0, both numerator and denominator are negative, making the fraction positive; so x<0 is NOT part of the solution set. \n" ); document.write( "For x values between 0 and 1, the numerator is positive and the denominator is negative, making the fraction negative; so x between 0 and 1 IS part of the solution set. Note the endpoint of that interval x=0 is part of the solution set, because it makes the numerator 0; the endpoint x=1 is NOT part of the solution set, because it makes the denominator 0. \n" ); document.write( "For x values greater than 1, both numerator and denominator are positive, making the fraction positive; so x>1 is NOT part of the solution set. \n" ); document.write( "And so the solution set -- as before, of course -- is [0,1) \n" ); document.write( " |