Question 1090691
.
((x-3)/(x+2)) > 0

thanks anyone.  
~~~~~~~~~~~~~~~~~


<pre>
There are 2 critical points x= 3 and x= -2,

and there are 3 intervals  ({{{-infinity}}},{{{-2}}}),  (-2,3)  and  ({{{3}}},{{{infinity}}}).


We consider all three intervals one after another, moving from the left to the right.


1.  In the interval  ({{{-infinity}}},{{{-2}}})  the numerator  (x-3) is negative; the denominator  (x+2) is negative, too.

    Hence, the ratio  {{{(x-3)/(x+2)}}}  is positive.



2.  In the interval  (-2,3)  the numerator  (x-3) is negative; the denominator  (x+2) is positive.

    Hence, the ratio  {{{(x-3)/(x+2)}}}  is negative.



3.  In the interval  ({{{3}}},{{{infinity}}})  the numerator  (x-3) is positive; the denominator  (x+2) is positive, too.

    Hence, the ratio  {{{(x-3)/(x+2)}}}  is positive.



4.  Thus your answer is:  The solution of the given inequality is this intervals union  ({{{-infinity}}},{{{-2}}})  U ({{{3}}},{{{infinity}}}).
</pre>

Solved.



The plot is below. You can consider it as visual confirmation to the solution.



{{{graph( 330, 330, -10.5, 10.5, -10.5, 10.5,
          (x-3)/(x+2)
)}}}


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Plot y = {{{(x-3)/(x+2)}}}



To see more solved problems/samples of this type, look into the lesson

&nbsp;&nbsp;&nbsp;&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> 

in this site.



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic &nbsp;"<U>Inequalities</U>".