1. **State the problem:** We have three vegetable patches with areas $w$, $2w + 5$, and $4w$ respectively. The total area is greater than 61 $m^2$. We need to write and solve an inequality for $w$. 2. **Write the inequality:** The total area is the sum of the three patches: $$w + (2w + 5) + 4w > 61$$ 3. **Simplify the inequality:** Combine like terms: $$w + 2w + 5 + 4w > 61$$ $$ (w + 2w + 4w) + 5 > 61$$ $$7w + 5 > 61$$ 4. **Isolate $w$:** Subtract 5 from both sides: $$7w + \cancel{5} - \cancel{5} > 61 - 5$$ $$7w > 56$$ 5. **Solve for $w$:** Divide both sides by 7: $$\frac{7w}{\cancel{7}} > \frac{56}{\cancel{7}}$$ $$w > 8$$ 6. **Interpretation:** The value of $w$ must be greater than 8 for the total area to be greater than 61 $m^2$. **Final answer:** $$w > 8$$