Explanation:
To solve the inequality \(x^3 - 9x^2 - 10x > 0\), you can follow these steps:
1. First, factor the expression:
\(x(x^2 - 9x - 10) > 0\)
2. Now, factor the quadratic expression inside the parentheses:
\(x(x - 10)(x + 1) > 0\)
3. Determine the critical points by setting each factor equal to zero and solving for \(x\):
\(x = 0\)
\(x - 10 = 0 \implies x = 10\)
\(x + 1 = 0 \implies x = -1\)
4. Create a sign chart using these critical points:
- On the left of -1, all factors are negative.
- Between -1 and 0, \(x\) is positive, but \((x - 10)\) and \((x + 1)\) are negative.
- Between 0 and 10, all factors are positive.
- To the right of 10, \(x\) and \((x - 10)\) are positive, but \((x + 1)\) is negative.
5. Now, determine the sign of the expression \(x(x - 10)(x + 1)\) in each interval:
- Left of -1: Negative * Negative * Negative = Negative
- Between -1 and 0: Positive * Negative * Negative = Positive
- Between 0 and 10: Positive * Positive * Positive = Positive
- Right of 10: Positive * Positive * Negative = Negative
6. The inequality \(x^3 - 9x^2 - 10x > 0\) is satisfied when the expression \(x(x - 10)(x + 1)\) is greater than zero, which occurs in the intervals where it is positive.
So, the solution to the inequality is:
\(x < -1\) or \(0 < x < 10\)
These are the values of \(x\) for which the original inequality is true.