√(4х-х^2)>-2-3х^2

To solve the inequality \( \sqrt{4x - x^2} > -2 - 3x^2 \), we can break this down step by step. 1. **First, simplify the inequality:** We have a square root, which is always non-negative, so we can ignore the right side if it's negative. Also, we need to ensure that the expression inside the square root is non-negative. From the square root \( \sqrt{4x - x^2} \), the expression inside must be greater than or equal to zero: \[ 4x - x^2 \geq 0 \] Factoring gives: \[ x(4 - x) \geq 0 \] This inequality holds true for \( x \) in the intervals \( [0, 4] \). 2. **Solve the inequality \( \sqrt{4x - x^2} > -2 - 3x^2 \):** Since the left side (\( \sqrt{4x - x^2} \)) is always non-negative, we focus on the right side \( -2 - 3x^2 \). For the entire inequality to hold, we need: \[ \sqrt{4x - x^2} \geq 0 > -2 - 3x^2 \] This tells us that \( -2 - 3x^2 < 0 \). Solving this gives: \[ -3x^2 < 2 \] or \[ 3x^2 > -2 \] Since \( 3x^2 \) is always non-negative for real \( x \), this inequality is always satisfied for all real \( x \). 3. **Combining the results:** Therefore, we only need to consider the range from \( 0 \) to \( 4 \) where the square root is defined: \( x \in [0, 4] \) 4. **Conclusion:** The solution for the inequality \( \sqrt{4x - x^2} > -2 - 3x^2 \) is: \[ x \in [0, 4] \] Thus, the solution to the inequality is \( x \in [0, 4] \).