Cherries cost 4 1 b. Grapes cost 2.50 Ib. You can spend no...

Cherries cost $$\$ 4$$ 1 b. Grapes cost $$\$ 2.50$$ $ Ib$. You can spend no more than $$\$ 15$$ on fruit, and you need
at least 4 lb in all. What is a graph showing the amount of each fruit you can buy?

Key Concepts

Decision Variables

Decision variables represent the quantities being determined in a problem. In the context of graphing problems, each variable corresponds to a specific quantity to be purchased or produced. Defining these variables clearly forms the basis for setting up equations or inequalities that model the situation.

Constraint Inequalities

Constraint inequalities are mathematical expressions that represent limitations or requirements in a problem. They are used to indicate restrictions such as budget limits, minimum or maximum quantities, or other conditions that must be satisfied. These inequalities form the boundaries of the solution set in the problem.

Graphing Linear Inequalities

Graphing linear inequalities involves plotting each inequality on a coordinate plane to visually represent the region of solutions that satisfy the inequality. This includes drawing the line associated with the equality, then shading the area of the plane that meets the inequality’s conditions, which helps in visualizing the feasible set.

Feasible Region

The feasible region is the intersection of all the shaded areas from the linear inequalities in a system. It represents all possible pairs of values for the decision variables that simultaneously satisfy all the problem constraints. This region is crucial in identifying potential solutions, especially in optimization problems, as it shows where every constraint is met.

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