Question 1004077
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solve 2x/9-x^2 > 0 
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{{{(2x)/(9-x^2)}}} > 0.



Write the left part in the form


{{{(2x)/((3-x)*(3+x))}}} > 0. 


It is equivalent to this inequality


{{{(2x)/((x-3)*(x+3))}}} < 0. &nbsp;&nbsp;&nbsp;&nbsp;(I changed the sign of one term in the denominator and changed the sign of the inequality).


Now at &nbsp;x < -3 &nbsp;the left part is negative. &nbsp;Do you understand why? 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Analyse the sigh of each factor in the numerator and denominator.


In the next interval, -3 < x < 0, the left part is positive. &nbsp;Do you understand why? 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Analyse the sigh of each factor in the numerator and denominator.


Move this way from left to right through the intervals &nbsp;(0, 3) &nbsp;and &nbsp;(3, {{{infinity}}}). &nbsp;Make the same analysis for each interval.


In this way you will find the final solution.


For similar problems see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Inequalities/Solving-inequalities-for-rat-functions-with-num-and-denom-factored-into-a-product-of-linear-binomials.lesson>Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials</A>&nbsp; in this site.


Please let me know if you succeed with this exercise.