3. Solve for \( x \) and write your solution in set-builder notation: ution: \[ 3<-3 x-6 \leq 5+\frac{x}{2} \]

To solve the compound inequality \( 3 < -3x - 6 \leq 5 + \frac{x}{2} \), we can break it down into two parts: 1. **From the first part \( 3 < -3x - 6 \)**, we simplify: \[ 3 + 6 < -3x \] \[ 9 < -3x \] Dividing by -3 (remember to flip the inequality): \[ -3 < x \] or \( x > -3 \). 2. **From the second part \( -3x - 6 \leq 5 + \frac{x}{2} \)**, we also simplify: \[ -3x - 6 \leq 5 + \frac{x}{2} \] Rearranging gives: \[ -3x - \frac{x}{2} \leq 5 + 6 \] To combine the \( x \) terms, multiply \( -3x \) by \( \frac{2}{2} \) to get: \[ \frac{-6x}{2} - \frac{x}{2} \leq 11 \] Combining gives: \[ \frac{-7x}{2} \leq 11 \] Multiplying by -2: \[ 7x \geq -22 \] Dividing by 7: \[ x \geq -\frac{22}{7} \]. Now, we combine the results: \[ -3 < x \leq -\frac{22}{7} \] In set-builder notation, the solution can be expressed as: \[ \{ x \mid -3 < x \leq -\frac{22}{7} \} \]