## Regions defined by inequality

It is possible to describe a set of points in a specific region through the use of inequalities. The solutions to inequalities can be represented as shaded regions on a graph, which sometimes include and sometimes exclude their boundaries.

Inequalities represent regions greater than (or greater & equal to) and less than (or less than & equal to). They can be as simple as y ≥ 2 or more complicated when involving a linear relation. In order to check whether the right regions are shaded, it is recommended to substitute various numbers into the equations to see if they remain true.

Sometimes more than one inequalities can be sketched on the same graph. The solution to both of them would be the overlapping region. This, in analysis, is referred to as the feasible region.

## Writing inequalities

In analysis, constraints are often given and then the feasible region is found. In order to find the solution to satisfy the objective function, one must first find the constraints by deducing from the information given.

### Application

Two students work together to produce a mathematics bound reference book consisting of text material and examples. Due to a tight schedule, the students have a maximum number of hours they are able to devote to this. For each chapter of text material and for each set of examples, the students devote the following amount of time (in hours) to the tasks:

Student A

• Text material: 20 hours
• Examples: 8 hours
• Maximum time available: 320 hours

Student B

• Text material: 10 hours
• Examples: 2 hours
• Maximum time available: 100 hours

If we let x = the number of chapters of text material and = the number chapers of examples, the constraints would be as follow:

$20x + 8y \leq 320$
which states that, FOR STUDENT A, total number of hours devoted to writing text materials and examples must be less than or equal to 320 hours.

$10x + 2y \leq 100$
which states that, FOR STUDENT B, the total number of hours devoted to writing text materials and examples must be less than or equal to 100 hours.