Solve the inequality (x - 1)/(x + 4) ≤ 0.

• A. x ≤ -4 or x ≥ 1
• B. x ∈ (-∞, -4] ∪ [1, ∞)
• C. x ∈ (-4, 1]
• D. x ∈ [-4, 1]

Answer: (C)

When solving a rational inequality, first find where the expression can be nonpositive by locating where the numerator is zero and where the denominator is zero, since those points determine potential zeros and the domain.

Here, the numerator x−1 is zero at x=1, giving value zero there (which satisfies ≤ 0). The denominator x+4 is zero at x=−4, which is not allowed (division by zero). So consider the intervals created by these points: (−∞, −4), (−4, 1], and (1, ∞).

Test a value from each interval:

• Pick x = −5: numerator negative, denominator negative, ratio positive (not allowed).

• Pick x = 0: numerator negative, denominator positive, ratio negative (allowed).

• Pick x = 2: numerator positive, denominator positive, ratio positive (not allowed).

Include x=1 because the value is exactly 0, which satisfies the inequality. Exclude x = −4 since it makes the expression undefined.

The solution is all x in the interval (−4, 1], i.e., numbers greater than −4 up to and including 1.