SOLUTION: Solve x-3/x-5 >= 0 for x and graph the solution on a number line.

 >= 0.


In this problem, there are two values of x, where the binomials in the numerator and denominator change their signs.


These values are x= 3  and  x= 5.


Notice that the domain of this inequality is the entire number line except of the value x= 5, where the denominator is zero.


The values (the points) 3 and 5 divide the number line (or, better to say, the domain) in three intervals:


    -infinity < x <=3;   3 <= x < 5   and  5 < x < infinity.


At the interval  -infinity < x <=3,  the numerator  (x-3) is negative; the denominator (x-5) is negative, too;  

hence, the fraction is positive.  Thus  the entire interval (,] is the part of the solution.



At the interval  3 <= x < 5,  the numerator  (x-3) is positive; the denominator (x-5) is negative;  

hence, the fraction is negative.  Thus  the entire interval [,) is NOT the part of the solution.



At the interval  5 < x < infinity,  the numerator  (x-3) is positive; the denominator (x-5) is positive;  

hence, the fraction is positive.  Thus  the entire interval (,) is the part of the solution.



ANSWER.  The solution to the given inequality is the union of two intervals   (,] U (,).