Inequalities

To solve the inequality , we need to follow these steps:

1. Identify the domain for the logarithm: The expression inside the logarithm, , must be positive since the logarithm of a non-positive number is undefined. Thus, we solve the inequality:

.

Factoring the quadratic, we get:

.

This inequality is satisfied when .

2. Solve the logarithmic inequality:

Convert the inequality from logarithmic form to exponential form:

Simplifying, we get:

.

Factor the quadratic:

.

This inequality holds for .

3. Combine the constraints: We need values of that satisfy both and . The intersection is:

.