Final answer:
The solution to the inequality x-1/3x+9 ≤ 0 can be found using critical points and testing intervals. The correct answer is option A.
Step-by-step explanation:
The solution to the inequality x - 1/(3x+9) ≤ 0 can be found by determining the critical points and testing the intervals between them. First, we set the numerator, x - 1, equal to zero:
x - 1 = 0
Solving this equation, we find that x = 1. We also need to consider the denominator, 3x + 9, and determine where it equals zero:
3x + 9 = 0
Solving for x, we get x = -3. Therefore, the critical points are x = -3 and x = 1. We can divide the number line into three intervals:
-∞ < x < -3, -3 < x < 1, and x > 1.
Now we test each interval by choosing a test value within it and plugging it into the inequality. For example, if we choose x = -4, plugging it into the inequality gives:
(-4) - 1/(3(-4)+9) ≤ 0
Simplifying this, we get -5/21 ≤ 0, which is true. We can repeat this process for the other intervals to determine the solution.
By testing the intervals, we find that the solution to the inequality is x < -3 or 1 ≤ x < ∞. Therefore, the correct answer is option A. (−3,−1)∪(1,[infinity])