x2 + 9x – 10 > 0

Sure, let's solve the inequality step by step:

1. First, we need to factor the quadratic equation. The equation x^2 + 9x - 10 = 0 factors to (x - 1)(x + 10) = 0.

2. Now, we find the roots of the equation by setting each factor equal to zero and solving for x. This gives us x = 1 and x = -10.

3. We then plot these points on a number line and test the intervals they create in the original inequality. The intervals are (-∞, -10), (-10, 1), and (1, ∞).

4. Let's test these intervals:

   • Choose x = -11 for (-∞, -10). Substituting -11 into the inequality gives (-11)^2 + 9(-11) - 10 > 0, which simplifies to 121 - 99 - 10 > 0, or 12 > 0. This is true, so (-∞, -10) is part of the solution.

   • Choose x = 0 for (-10, 1). Substituting 0 into the inequality gives 0^2 + 9(0) - 10 > 0, which simplifies to -10 > 0. This is false, so (-10, 1) is not part of the solution.

   • Choose x = 2 for (1, ∞). Substituting 2 into the inequality gives 2^2 + 9(2) - 10 > 0, which simplifies to 4 + 18 - 10 > 0, or 12 > 0. This is true, so (1, ∞) is part of the solution.

5. Therefore, the solution to the inequality x^2 + 9x - 10 > 0 is x ∈ (-∞, -10) U (1, ∞).