Question 1133221: Solve x - 3/x - 5 >= 0 and graph the solution on a number line
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
I will assume the inequality is in fact (x-3)/(x-5) >= 0....
At x=3, the numerator is zero, so the function value is 0.
At x=5, the denominator is zero, so the function is undefined.
Those two critical values of x divide the number line into three segments: -infinity to 3, 3 to 5, and 5 to infinity. Since x=3 anad x=5 are the only values of x for which the denominator is either 0 or undefined, the function always has either a positive or negative value on each of those intervals.
(-infinity,3): numerator and denominator both negative, function value positive
(3,5): numerator positive, denominator negative, function value negative
(5,infinity): numerator and denominator both positive, function value positive
The inequality is for greater than or equal to 0. Since the function value is 0 at 3 and undefined at 5, the solution set for the inequality is
(-infinity,3] U (5,infinity)
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