(1) \( x>-4 \) and \( x \leqslant-1 \) (2) \( x+5>4 \) and \( x-2<2 \) (3) \( y \leqslant-2 \) or \( y>3 \) (4) \( -3 t>12 \) or \( 5 t \geqslant 10 \)

Let's solve each of the given inequalities step by step. --- ### 1. \( x > -4 \) **and** \( x \leqslant -1 \) **Solution:** To satisfy both conditions simultaneously, \( x \) must be greater than \(-4\) **and** less than or equal to \(-1\). \[ -4 < x \leq -1 \] **Solution Set:** \[ x \in (-4,\, -1] \] --- ### 2. \( x + 5 > 4 \) **and** \( x - 2 < 2 \) **Solution:** 1. **First Inequality:** \[ x + 5 > 4 \\ \Rightarrow x > 4 - 5 \\ \Rightarrow x > -1 \] 2. **Second Inequality:** \[ x - 2 < 2 \\ \Rightarrow x < 2 + 2 \\ \Rightarrow x < 4 \] **Combined Solution:** \[ -1 < x < 4 \] **Solution Set:** \[ x \in (-1,\, 4) \] --- ### 3. \( y \leqslant -2 \) **or** \( y > 3 \) **Solution:** This inequality represents two separate intervals where either condition can be true. 1. **First Condition:** \[ y \leq -2 \] 2. **Second Condition:** \[ y > 3 \] **Solution Set:** \[ y \in (-\infty,\, -2] \cup (3,\, \infty) \] --- ### 4. \( -3t > 12 \) **or** \( 5t \geqslant 10 \) **Solution:** 1. **First Inequality:** \[ -3t > 12 \\ \Rightarrow t < \frac{12}{-3} \quad \text{(Divide by -3, inequality sign reverses)} \\ \Rightarrow t < -4 \] 2. **Second Inequality:** \[ 5t \geq 10 \\ \Rightarrow t \geq \frac{10}{5} \\ \Rightarrow t \geq 2 \] **Solution Set:** \[ t \in (-\infty,\, -4) \cup [2,\, \infty) \] --- ### Summary of Solutions 1. \( x \in (-4,\, -1] \) 2. \( x \in (-1,\, 4) \) 3. \( y \in (-\infty,\, -2] \cup (3,\, \infty) \) 4. \( t \in (-\infty,\, -4) \cup [2,\, \infty) \) If you have any further questions or need additional explanations, feel free to ask!