Final answer:
To find the solution to the inequality (x-5)/(x²)(x-4) > 0, examine critical points and test intervals to determine where the function is positive. The solution is 'x > 5 or x < 4'.
Step-by-step explanation:
To solve the inequality (x-5)/(x²)(x-4) > 0, we must consider the critical points of the function, which occur where the numerator or denominator is zero. These points are x = 5 (numerator becomes zero) and x = 0, x = 4 (denominator becomes zero). Since the denominator includes a square term x², it is always positive except for x = 0 where it is undefined. Thus, we only need to consider the change of sign caused by (x-5) and (x-4). We can analyze the intervals determined by our critical points by creating a sign chart or test values from each interval to determine where the function is positive.
By testing values to the left of 4, between 4 and 5, and to the right of 5, we find that the function is positive when x < 4 and when x > 5. Therefore, the solution of the inequality is x > 5 or x < 4, which corresponds to option (a).