\( -2 x-7>1 \) or \( x-2 \geq-1 \) Solution

To solve the inequalities \( -2x - 7 > 1 \) **or** \( x - 2 \geq -1 \), we'll solve each inequality separately and then combine their solutions. ### 1. Solving \( -2x - 7 > 1 \) **Step 1:** Add 7 to both sides to isolate the term with \( x \): \[ -2x - 7 + 7 > 1 + 7 \\ -2x > 8 \] **Step 2:** Divide both sides by -2. **Remember:** Dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign. \[ x < \frac{8}{-2} \\ x < -4 \] **Solution for the first inequality:** \[ x < -4 \] ### 2. Solving \( x - 2 \geq -1 \) **Step 1:** Add 2 to both sides to solve for \( x \): \[ x - 2 + 2 \geq -1 + 2 \\ x \geq 1 \] **Solution for the second inequality:** \[ x \geq 1 \] ### Combining the Solutions Since the original problem uses **"or"**, the final solution includes all \( x \) that satisfy **either** of the two inequalities. \[ x < -4 \quad \text{**or**} \quad x \geq 1 \] **In interval notation:** \[ (-\infty, -4) \cup [1, \infty) \] ### Final Answer All real numbers below –4 or at least 1. In symbols, x < −4 ∨ x ≥ 1 Or in interval notation: (−∞, −4) ∪ [1, ∞)