Double Inequality A35A41

1. **State the problem:** Solve the double inequality $$-6.5 \leq \frac{7x + 6}{2} \leq 20.5$$ and find any three solutions. 2. **Write the inequality:** $$-6.5 \leq \frac{7x + 6}{2} \leq 20.5$$ 3. **Multiply all parts by 2 to eliminate the denominator:** $$2 \times -6.5 \leq 7x + 6 \leq 2 \times 20.5$$ $$-13 \leq 7x + 6 \leq 41$$ 4. **Subtract 6 from all parts:** $$-13 - 6 \leq 7x + 6 - 6 \leq 41 - 6$$ $$-19 \leq 7x \leq 35$$ 5. **Divide all parts by 7:** $$\frac{-19}{7} \leq x \leq \frac{35}{7}$$ 6. **Simplify fractions:** $$-\frac{19}{7} \leq x \leq 5$$ 7. **Interpretation:** The solution set is all real numbers $$x$$ such that $$-\frac{19}{7} \leq x \leq 5$$. 8. **Choose any three solutions within the interval:** $$x = -2, 0, 4$$ **Final answer:** $$-\frac{19}{7} \leq x \leq 5$$ with example solutions $$x = -2, 0, 4$$.