D. \( m \geq 6.3 \) Which solutions do the inequalities \( \frac{1}{2}(5 r+3) \geq 14 \) and \( -2 s+6 \geq-8 \) have in common? Show your work.

To find the common solutions of the inequalities \( \frac{1}{2}(5r + 3) \geq 14 \) and \( -2s + 6 \geq -8 \), we will solve each inequality step by step. ### Step 1: Solve the first inequality \( \frac{1}{2}(5r + 3) \geq 14 \) 1. **Multiply both sides by 2** to eliminate the fraction: \[ 5r + 3 \geq 28 \] 2. **Subtract 3 from both sides**: \[ 5r \geq 25 \] 3. **Divide both sides by 5**: \[ r \geq 5 \] ### Step 2: Solve the second inequality \( -2s + 6 \geq -8 \) 1. **Subtract 6 from both sides**: \[ -2s \geq -14 \] 2. **Divide both sides by -2** (remember to reverse the inequality sign): \[ s \leq 7 \] ### Step 3: Determine the common solutions Now we have the solutions: - For \( r \): \( r \geq 5 \) - For \( s \): \( s \leq 7 \) Since the inequalities involve different variables \( r \) and \( s \), we cannot find a common solution in the traditional sense. However, we can express the solution set for each variable: - The solution for \( r \) is \( r \in [5, \infty) \). - The solution for \( s \) is \( s \in (-\infty, 7] \). ### Conclusion The inequalities do not have a common solution in terms of a single variable, but the solution sets are: - \( r \geq 5 \) - \( s \leq 7 \) These represent the ranges of values for \( r \) and \( s \) that satisfy their respective inequalities.