Answer: To solve the rational inequality:
x+1 / x−3 ≤ 0
We can start by finding the critical points of the inequality by setting the numerator and denominator equal to zero:
x + 1 = 0 => x = -1
x - 3 = 0 => x = 3
These critical points divide the number line into three intervals: (-infinity, -1), (-1, 3), and (3, infinity). We can then test each interval to determine whether the inequality is true or false.
Testing the interval (-infinity, -1), we choose a test point x = -2:
(-2 + 1) / (-2 - 3) = -1/5 < 0
Therefore, this interval satisfies the inequality.
Testing the interval (-1, 3), we choose a test point x = 0:
(0 + 1) / (0 - 3) = -1/3 > 0
Therefore, this interval does not satisfy the inequality.
Testing the interval (3, infinity), we choose a test point x = 4:
(4 + 1) / (4 - 3) = 5 > 0
Therefore, this interval satisfies the inequality.
So, the solution to the inequality is:
x ∈ (-∞, -1] ∪ (3, ∞)
In interval notation, this solution is (-∞, -1] U (3, ∞).
Explanation: