Find the solution set of the inequality (x^2 - 3x - 10) / (x - 4) ≤ 0.

Correct answer

B. (-∞, -2] ∪ (4, 5]

Principle or equation

For a rational inequality, find critical points where numerator or denominator is zero, then test intervals on the real number line. The denominator cannot be zero.

Why this answer is correct

Factor the numerator: x^2 - 3x - 10 = (x - 5)(x + 2). Critical points are x = -2, x = 5 (numerator zero) and x = 4 (denominator zero). These divide the real line into intervals: (-∞, -2), (-2, 4), (4, 5), (5, ∞). Test each interval: for x = -3, expression is positive; for x = 0, expression is positive? Actually compute: at x=0, (0-5)(0+2)/(0-4)=(-5*2)/(-4)=10/4=2.5>0; at x=4.5, (4.5-5)(4.5+2)/(4.5-4)=(-0.5*6.5)/0.5=-6.50. Also include endpoints where numerator is zero: x=-2 and x=5 make expression zero, so they are included. x=4 is excluded. Thus solution: (-∞, -2] ∪ (4, 5].

Example

Solve (x^2 - 1)/(x - 2) ≤ 0. Factor numerator: (x-1)(x+1). Critical: -1,1,2. Test intervals: (-∞,-1] positive? Actually at -2: (-3*-1)/(-4)=3/-4=-0.75 ≤0, so include; (-1,1) positive? at 0: (-1*1)/(-2)=0.5>0; (1,2) negative? at 1.5:0.5*2.5/-0.5=-2.5 ≤0; (2,∞) positive. So solution: (-∞,-1] ∪ [1,2).