Solve the inequality (x+2)/(x-4) > 0.

• A. (-infinity, -2) ∪ (4, infinity)
• B. (-infinity, -4) ∪ (2, infinity)
• C. (-infinity, -2] ∪ [4, infinity)
• D. (-infinity, -2) ∪ [4, infinity)

Answer: (A)

To solve a rational inequality, focus on where the expression can change sign by looking at where the numerator or denominator is zero. For (x+2)/(x-4), the critical points are x = -2 (where the numerator is zero) and x = 4 (where the denominator is zero, meaning the expression is undefined). Because the inequality is strict (> 0), we must exclude x = -2 and x = 4 from the solution.

Divide the real line into intervals: (-∞, -2), (-2, 4), and (4, ∞). Test a representative value in each interval:

• In (-∞, -2), take x = -3: numerator is -1 (negative) and denominator is -7 (negative), so the ratio is positive. The inequality holds here.

• In (-2, 4), take x = 0: numerator is 2 (positive) and denominator is -4 (negative), so the ratio is negative. The inequality does not hold.

• In (4, ∞), take x = 5: numerator is 7 (positive) and denominator is 1 (positive), so the ratio is positive. The inequality holds here.

Thus, the solution is all x less than -2 or greater than 4, with -2 and 4 excluded: (-∞, -2) ∪ (4, ∞).