Final answer:
The inequality (x - 5)^2(x - 2)(x - 1) > 0 is solved by finding the zeros and testing intervals, ultimately expressed in interval notation as (-∞, 1) ∪ (2, 5) ∪ (5, ∞).
Step-by-step explanation:
To solve the inequality (x - 5)^2(x - 2)(x - 1) > 0, we will first identify the zeros of the function, which occur at x=1, x=2, and x=5. Note that the zero at x=5 is a double zero because of the squared term. We then determine the signs on each interval determined by the zeros. The intervals are (-∞, 1), (1, 2), (2, 5), and (5, ∞). Testing points from each interval:
- For x in (-∞, 1), choose x=0: (-5)^2(-2)(-1) = 10 > 0.
- For x in (1, 2), choose x=1.5: (-3.5)^2(-0.5)(0.5) = -3.0625 < 0.
- For x in (2, 5), choose x=3: (-2)^2(1)(2) = 4 > 0.
- For x in (5, ∞), choose x=6: (1)^2(4)(5) = 20 > 0.
We include only the intervals where the product is positive. Taking into account the end behavior where x approaches positivity, we express the solution in interval notation as: (-infty, 1) ∪ (2, 5) ∪ (5, ∞).