Solve the inequality (x^2 + 3x - 10) / (x - 1) < 0. Which of the following is the solution set?

Correct answer

B. (-∞, -5) ∪ (1, 2)

Principle or equation

To solve a rational inequality, find critical points where numerator or denominator is zero, then test intervals on a number line.

Why this answer is correct

Factor numerator: x^2 + 3x - 10 = (x + 5)(x - 2). Critical points: x = -5, 2 (from numerator) and x = 1 (from denominator). These divide the real line into intervals: (-∞, -5), (-5, 1), (1, 2), (2, ∞). Test a point in each interval: for x = -6, expression is positive; for x = 0, expression is positive? Actually compute: (0+5)(0-2)/(0-1)=5*(-2)/(-1)=10 >0, so positive. For x = 1.5, (1.5+5)(1.5-2)/(1.5-1)=6.5*(-0.5)/0.5=-6.5 0. So inequality holds on (1,2). Also need to exclude x=1 because denominator zero. So solution is (-5,1) ∪ (1,2)? Wait check interval (-5,1): test x=0 gave positive, so not included. So only (1,2). But options do not have just (1,2). Let's re-evaluate. Actually test x=0: (0+5)(0-2)/(0-1)=5*(-2)/(-1)=10 >0, so positive. So negative only on (1,2). So solution is (1,2). But none of the options match. Did I miscompute? Let's recalc: (x^2+3x-10)/(x-1) = (x+5)(x-2)/(x-1). Critical points: -5, 1, 2. Sign chart: For x < -5, e.g., -6: (-1)(-8)/(-7) = 8/ -7? Actually (-1)*(-8)=8, divided by -7 = -8/7? Wait denominator x-1 = -7, so 8/(-7) = -8/7 negative. So negative on (-∞,-5). For -5 < x < 1, e.g., 0: (5)(-2)/(-1)=10 positive. For 1 < x < 2, e.g., 1.5: (6.5)(-0.5)/0.5 = -6.5 negative. For x > 2, e.g., 3: (8)(1)/2=4 positive. So negative on (-∞,-5) and (1,2). So solution is (-∞,-5) ∪ (1,2). That matches option 2. I made a sign error earlier. Good.

Example

Solve (x^2 - 1)/(x - 2) < 0. Critical points: -1, 1, 2. Test intervals: negative on (-∞,-1) and (1,2). Solution: (-∞,-1) ∪ (1,2).