Final answer:
The solution set for the inequality (x-5)(x-3) ≤ 0 is OPTION.C. [3,5], as this is the interval where the product of the two factors is either zero or negative.
Step-by-step explanation:
The student is asking about the solution set of the inequality (x-5)(x-3) ≤ 0. To find this solution set, we can analyze the signs of the product in various intervals that are determined by the roots of the equation (x-5)(x-3) = 0, which are x = 3 and x = 5. The inequality will hold wherever the product of the two factors is negative or zero.
Since both factors are linear, and their product changes sign at the roots, we divide the x-axis into three intervals: (-∞, 3), (3, 5), (5, +∞). Testing points from each interval in the inequality, we find that the inequality is true for x in the interval between 3 and 5. Additionally, since the inequality is non-strict (includes equals), x = 3 and x = 5 also satisfy it. Hence, the correct solution set is [3,5].