Final answer:
The solution set to the inequality 5(x - 2)(x + 4) > 0 is x < -4 and x > 2, which are the intervals where the product of the factors and the constant is positive.
Step-by-step explanation:
To solve the inequality 5(x – 2)(x + 4) > 0, we need to find the values of x that make the expression positive. First, identify the critical points by setting the inequality to zero: x – 2 = 0 and x + 4 = 0. Solving these, we get the critical points x = 2 and x = -4. Since the inequality is a product of two linear factors and a positive constant, the sign of the product changes at these critical points.
To determine the sign of the product in each interval, we can test points in the intervals (-∞, -4), (-4, 2), and (2, ∞). Testing these intervals, we find that the product is positive when x is in the intervals (-∞, -4) and (2, ∞). Therefore, the solution set to the inequality is x < -4 and x > 2.