Algebra.Com's Answer #774504 by ikleyn(52780)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
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document.write( "Below is the plot of two functions, y =  (in red) and y = 3x (in green).\r
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document.write( "The problem is to find where the red line is above the green line.\r
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document.write( " Plot y = (red) and y = 3x (green).\r\n" );
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document.write( "From the plot, it is clear that the solution is the union of TWO intervals:\r
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document.write( " one interval is semi-infinite ( , ) and the other interval is somewhere between -1 and 0.\r
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document.write( "The solution by @MathLover1 gives only the second interval and entirely misses the first semi-infinite interval.\r
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document.write( "So, her solution is INCORRECT. Therefore, I came to bring the correct solution.\r
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document.write( "After completing my solution, I will point the error in the MathLover1' solution.\r
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document.write( "So, we start from the given inequality\r\n" );
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document.write( " > 3x. (1)\r\n" );
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document.write( "Move 3x to the left side\r\n" );
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document.write( " - 3x > 0\r\n" );
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document.write( "Write with the common denominator\r\n" );
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document.write( " -  > 0\r\n" );
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document.write( "Make equivalent transformations\r\n" );
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document.write( "  > 0\r\n" );
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document.write( "  > 0\r\n" );
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document.write( "  > 0\r\n" );
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document.write( "  > 0\r\n" );
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document.write( "Multiply both sides by (-1) and change the inequality sign\r\n" );
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document.write( "  < 0.\r\n" );
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document.write( "Divide by 3 both sides\r\n" );
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document.write( "  < 0. (2)\r\n" );
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document.write( "So, our original inequality is equivalent to the inequality (2).\r\n" );
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document.write( "Now I will solve the inequality (2).\r\n" );
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document.write( "The factors in inequality (2) have three critical points, where linear terms become equal to zero and change the sign.\r\n" );
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document.write( "These points are -1,  and 0.\r\n" );
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document.write( "The critical points divide the entire number line in four intervals\r\n" );
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document.write( " 1) (,), 2) (,), 3) (, ) and 4) [, ).\r\n" );
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document.write( "In the interval #1, all three linear factors are negative, so the entire rational function is negative.\r\n" );
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document.write( "Thus the entire interval (,) IS the solution to inequality (2) and, hence, for the original inequality, too.\r\n" );
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document.write( "In the interval #2, the linear factor (x+1) is positive, while two other factors are negative, \r\n" );
document.write( "so the entire rational function is positive.\r\n" );
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document.write( "Thus the interval (,) is NOT the solution to inequality (2) and, hence, is NOT the solution for the original inequality.\r\n" );
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document.write( "In the interval #3, the linear factors (x+1) and  are both positive, while third factor x is negative, \r\n" );
document.write( "so the entire rational function is negative.\r\n" );
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document.write( "Thus the interval (,) IS the solution to inequality (2) and, hence, IS the solution for the original inequality.\r\n" );
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document.write( "In the interval #4, all three linear factors are positive, so the entire rational function in (2) is positive.\r\n" );
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document.write( "Thus the interval [,) is NOT the solution to inequality (2) and, hence, is not the solution for the original inequality.\r\n" );
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document.write( "ANSWER. The solution set for the given inequality is the union of two intervals (,) U (,).\r\n" );
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document.write( "Please notice and pay special attention that all endpoints are treated correctly in my solution and in my answer.\r\n" );
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document.write( "The problem is solved, and solved correctly.\r
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document.write( "The way on how I solved it, is a STANDARD (but not a unique) way solving such inequalities for rational functions.\r
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document.write( "Now, I promised to show where is the error in the solution by @MathLover1.\r
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document.write( "It is in the 4-th line of her post, where she multiplies both sides of the inequality by the factor (x+1)\r
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document.write( "without analyzing the sign of this factor.\r
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document.write( "Doing in this way, she gets NON-EQUIVALENT inequality and comes to the wrong resulting answer.\r
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document.write( "It is very common error in solving such inequalities for rational functions - so, be aware (!)\r
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document.write( "At this point, I completed my post.\r
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document.write( "From it, learn on how to solve similar problems correctly.\r
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document.write( "-----------------\r
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document.write( "If you want to see many other similar solved problems, look into the lessons\r
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document.write( " - Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials \r
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document.write( " - Solving inequalities for rational functions with non-zero right side \r
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document.write( "in this site.\r
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document.write( "Consider these lessons as your handbook, textbook, tutorials and (free of charge) home teacher.\r
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