\( \frac{a}{-2}<-1 \) or \( -4 a+3 \geq 23 \) Solution Interval Notation \( \square \)

Let's solve the compound inequality step by step and express the solution in interval notation. **Given Inequalities:** \[ \frac{a}{-2} < -1 \quad \text{or} \quad -4a + 3 \geq 23 \] ### 1. Solve the first inequality: \(\frac{a}{-2} < -1\) \[ \frac{a}{-2} < -1 \] **Step 1:** Multiply both sides by \(-2\). **Note:** Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. \[ a > 2 \quad \text{(Inequality sign reverses)} \] ### 2. Solve the second inequality: \(-4a + 3 \geq 23\) \[ -4a + 3 \geq 23 \] **Step 1:** Subtract 3 from both sides. \[ -4a \geq 20 \] **Step 2:** Divide both sides by \(-4\). **Note:** Dividing by a negative number reverses the inequality sign. \[ a \leq -5 \quad \text{(Inequality sign reverses)} \] ### 3. Combine the Solutions Since the original inequalities are connected by "or," the solution includes all values that satisfy **either** inequality. \[ a \leq -5 \quad \text{or} \quad a > 2 \] ### 4. Express in Interval Notation \[ (-\infty, -5] \cup (2, \infty) \] **Final Answer:** \[ \boxed{\,(-\infty,\ -5\,]\,\cup\, (2,\ \infty)\,} \]