A community garden is planning a rectangular planting area. Let $t$ be the number of tomato plants and $p$ the number of pepper plants. Each tomato plant requires $2{\text{ft}}^2$ of space and each pepper plant requires $1.5{\text{ft}}^2$. The total available area is at most $120{\text{ft}}^2$, and at least $10$ tomato plants and $8$ pepper plants must be planted. Form a system of inequalities that models these conditions. Then determine all integer pairs $(t,p)$ that satisfy the area and minimum planting constraints, assuming $t$ and $p$ are nonnegative integers.