Final answer:
To solve the inequality x^3 - 4x^2 - 45x > 0, factor out an x to obtain x(x - 9)(x + 5) > 0. Find the critical points, which are x = 0, x = 9, and x = -5, and use them to determine the intervals where the expression is positive. The solution is x in the intervals (-5, 0) and (9, Infinity).
Step-by-step explanation:
To solve the inequality x^3 - 4x^2 - 45x > 0, we'll first factor the left side of the inequality. Since all terms have an 'x' in common, we can factor out an x:
x(x^2 - 4x - 45) > 0
Next, we'll find factors of the quadratic that multiply to give -45 and add to give -4. The factors that meet these criteria are -9 and 5, so we can factor the quadratic as:
x(x - 9)(x + 5) > 0
Now, we have the inequality in factored form, which allows us to find the critical points of x that will help us determine the intervals where the inequality holds true. The critical points are x = 0, x = 9, and x = -5.
Determining the sign of each interval between and beyond these critical points, we can use a sign chart or test values within the intervals. As a result, we find that positive intervals are:
The solution to the inequality is x in the interval (-5, 0) U (9, Infinity). This is where the original expression is positive.