Решение

To solve the inequality \( \log_{x+2}(x^2 - 2) \geq 2 \), follow these steps: 1. **Understand the Domain:** - The base \( x+2 \) must be positive and not equal to 1: \( x+2 > 0 \) and \( x+2 \neq 1 \). Thus, \( x > -2 \) and \( x \neq -1 \). - The argument \( x^2 - 2 \) must be positive: \( x^2 - 2 > 0 \). Thus, \( x < -\sqrt{2} \) or \( x > \sqrt{2} \). 2. **Rewrite the Inequality:** \[ \log_{x+2}(x^2 - 2) \geq 2 \quad \Rightarrow \quad x^2 - 2 \geq (x+2)^2 \] 3. **Simplify:** \[ x^2 - 2 \geq (x^2 + 4x + 4) \] \[ -2 \geq 4x + 4 \] \[ -6 \geq 4x \] \[ x \leq -\frac{3}{2} \] 4. **Find the Intersection of Conditions:** - From domain: \( x > -2 \) or \( x \leq -\frac{3}{2} \) - Intersection is \( -2 < x \leq -\frac{3}{2} \). 5. **Solution:** - The solution to the inequality is \( -2 < x \leq -\frac{3}{2} \).