1. **State the problem:** Solve the inequality $$4 \geq -7j - 6 \geq -10$$ for the variable $j$. 2. **Rewrite the compound inequality:** This means we have two inequalities to solve simultaneously: $$4 \geq -7j - 6$$ and $$-7j - 6 \geq -10$$ 3. **Solve the first inequality:** $$4 \geq -7j - 6$$ Add 6 to both sides: $$4 + 6 \geq -7j - 6 + 6$$ $$10 \geq -7j$$ Divide both sides by $-7$ (note: dividing by a negative number reverses the inequality): $$\frac{10}{\cancel{-7}} \leq \frac{-7j}{\cancel{-7}}$$ $$-\frac{10}{7} \leq j$$ 4. **Solve the second inequality:** $$-7j - 6 \geq -10$$ Add 6 to both sides: $$-7j - 6 + 6 \geq -10 + 6$$ $$-7j \geq -4$$ Divide both sides by $-7$ (again reversing the inequality): $$\frac{-7j}{\cancel{-7}} \leq \frac{-4}{\cancel{-7}}$$ $$j \leq \frac{4}{7}$$ 5. **Combine the results:** $$-\frac{10}{7} \leq j \leq \frac{4}{7}$$ **Final answer:** $$j \in \left[-\frac{10}{7}, \frac{4}{7}\right]$$