Question 62776
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help please
solve the rational inequality (x-2)/(x+3) is less than or equal to 0
is it: (-3,2] (-oo,-3) (-oo,-3)U{2,oo) or [2,oo)
i am very confused
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        The solution in the post by @jai_kos in incorrect.

        It is incorrect methodologically and gives incorrect answer.

        See my correct solution below.



<pre>
They want you solve this inequality

    {{{(x-2)/(x+3)}}} <= 0.    (1)


The left side rational function can be non-positive if and only if

    EITHER the numerator is non-positive and denominator is positive

        x - 2 <= 0  and  x + 3 > 0    (2)

    OR     the numerator is non-negative and denominator is negative

        x - 2 >= 0  and  x + 3 < 0.   (3)



In case (2),  x <= 2  and  x > -3  simultaneously, or  

              -3 < x <= 2.



In case (3),  x >= 2  and  x < -3 simultaneously,  which has no solutions.  



Thus the final solution to the given inequality is this set of real numbers  -3 < x <= 2,
or, in the interval notation, the set (-3,2].
</pre>

Solved.


The error made by &nbsp;@jai_kos is that when he multiplies both sides of the original inequality by &nbsp;(x+3),
he misses the case when &nbsp;(x+3) &nbsp;is negative, &nbsp;which requires different treatment.


This error, &nbsp;which jai_kos makes solving the problem, &nbsp;is a typical error, &nbsp;which beginners make 
when trying to solve such inequalities,
until the more experienced teachers/tutors will explain their error and will show a right way solving.