Final answer:
To solve the inequality (3x-5)/(x-3) ≤ 1, we find the critical points and test intervals, which shows that the solution is x ∈ (-∞, 1] ∪ (1, 3). The correct answer in interval notation is (-∞, 1], as x cannot equal 3.
Step-by-step explanation:
To solve the inequality (3x-5)/(x-3) ≤ 1, we need to find the values of x that satisfy this condition. We can start by bringing all terms to one side of the inequality to have a common denominator:
(3x - 5)/(x - 3) - 1 ≤ 0
Now combine the terms:
(3x - 5 - (x - 3))/(x - 3) ≤ 0
This simplifies to:
(2x - 2)/(x - 3) ≤ 0
Further simplification yields:
2(x - 1)/(x - 3) ≤ 0
We have to consider the critical points, which are x = 1 and x = 3, the points where the numerator and denominator change signs. These points divide the number line into different intervals. Testing points from each interval will help determine where the inequality holds true. It's important to note that x cannot equal 3 because it would make the denominator zero, which is undefined.
Upon testing these intervals, we find that the solution to the inequality is x ∈ (-∞, 1] ∪ (1, 3).
This encompasses both the solutions a) (-∞, 1] and b) [1, 3) from the original question, but since x cannot equal 3, we exclude this point. Therefore, the correct answer is a) (-∞, 1].