Which boundary type is drawn for the inequality y < 2x + 1 and why?

Put the inequality 2x + 5y ≤ 10 into slope-intercept form, identify the y-intercept and slope, and state whether the boundary is solid or dashed.

Given the system y ≤ x - 2 and y > x - 1, evaluate whether the system has any solutions and justify the reasoning.

A company makes two products: $A$ and $B$. Each product $A$ gives \(30 in profit and uses 2 hours of labor; each product $B$ gives \)50 in profit and uses 3 hours of labor. The weekly labor available is at most 60 hours per week. They must make at least 5 units of product $A$. There is no minimum for $B$. Set up a system of inequalities and determine which corner of the feasible region (integer or fractional that is allowed) that will maximizes the profit: $P=30A+50B$. Show all calculations.

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.