- Define variables: Let r = number of rabbits, d = number of dogs.
- Inequality 1: r + d ≤ 20 (combined total never more than 20).
- Inequality 2: r ≤ 8 (never more than 8 rabbits).
- Non-negativity: r ≥ 0, d ≥ 0.
- The solution region is the area on the graph bounded by these four inequalities.
- Any point (r, d) within or on the boundary of this region represents a valid combination of rabbits and dogs.
- For example, (5, 10) is a solution: 5 rabbits and 10 dogs (total 15 ≤ 20, rabbits 5 ≤ 8).
- Another example: (8, 12) is a solution: 8 rabbits and 12 dogs (total 20 ≤ 20, rabbits 8 ≤ 8).
- The solution is not a single point but all integer coordinate points in the feasible region.
- Define variables: Let w = number of wing packages, h = number of hot dog packages.
- Part a: 7w + 5h ≤ 40 (total cost no more than $40).
- Part b: h ≥ 5 (at least 5 packages of hot dogs).
- Non-negativity: w ≥ 0, h ≥ 0.
- Graph both inequalities: The line 7w + 5h = 40 intersects axes at (0, 8) and (40/7 ≈ 5.71, 0). Shade below it. The line h = 5 is horizontal; shade above it.
- The solution region is the overlap of shaded areas, bounded by h=5, 7w+5h=40, w=0, and h≥5.
- Find two points in the solution region:
- Point 1: (0, 5) — Buy 0 wings, 5 hot dogs. Cost = 7(0) + 5(5) = $25 ≤ $40. Valid.
- Point 2: (2, 5) — Buy 2 wings, 5 hot dogs. Cost = 7(2) + 5(5) = 14 + 25 = $39 ≤ $40. Valid.
- Other valid points include (1,5), (3,5), (0,6), (1,6), etc., as long as they satisfy both inequalities.
Parent Tip: Review the logic above to help your child master the concept of system of inequalities word problems worksheet.
