Set up the necessary inequalities and sketch the graph of...

Set up the necessary inequalities and sketch the graph of the region in which the points satisfy the indicated inequality or system of inequalities. A telephone company is installing two types of fiber-optic cable in an area. It is estimated that no more than 300 m of type A cable, and at least 200 m but no more than 400 m of type $B$ cable, are needed. Graph the possible lengths of cable that are needed.

Set up the necessary inequalities and sketch the graph of the region in which the points satisfy the indicated inequality or system of inequalities.
A telephone company is installing two types of fiber-optic cable in an area. It is estimated that no more than 300 m of type A cable, and at least 200 m but no more than 400 m of type $B$ cable, are needed. Graph the possible lengths of cable that are needed.

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Key Concepts

Inequalities

An inequality is a mathematical statement that relates two expressions using symbols like <, ?, >, or ?. In real-world applications, they are used to define limits such as minimum or maximum allowable quantities.

System of Inequalities

A system of inequalities consists of multiple inequalities that must all be satisfied simultaneously. The solution to such a system is the intersection of the regions defined by each individual inequality, representing all values that meet every condition.

Graphing Inequalities

Graphing inequalities involves plotting the boundary lines that mark the limits of the inequality and then shading the appropriate side of the line to represent all the values that satisfy the inequality. This visual method helps in identifying the region of solutions.

Feasible Region

The feasible region is the common area where all the inequalities in a system overlap. This region contains all the possible solution points that meet all the specified constraints.

Bounded Constraints

Bounded constraints specify the upper and lower limits for a variable. They are essential in modeling real-world scenarios where resources are limited or requirements have specific minimum or maximum values.

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