System of Inequalities

Definition

A system of inequalities is a collection of two or more inequalities that share the same variables. These inequalities represent constraints that define a feasible region in a coordinate plane, where any point within this region satisfies all the inequalities simultaneously. The solutions to a system of inequalities are often represented graphically, highlighting the areas where the constraints overlap, which is essential in optimization problems.

5 Must Know Facts For Your Next Test

1. When graphing a system of inequalities, each inequality divides the coordinate plane into two regions, and the solution is found where these regions intersect.
2. If an inequality is strict (using < or >), the boundary line is dashed, indicating that points on the line are not included in the solution set; if it's non-strict (using ≤ or ≥), the line is solid.
3. Systems of inequalities can be solved using graphical methods or algebraic methods such as substitution or elimination.
4. In optimization problems, systems of inequalities define constraints under which a certain objective function is maximized or minimized.
5. The vertices of the feasible region are crucial because they can often be evaluated to find the optimal solution for linear programming problems.

Review Questions

• How do you graph a system of inequalities and identify the feasible region?
  • To graph a system of inequalities, start by converting each inequality into an equation to find the boundary lines. Then graph each line on the coordinate plane. Use a dashed line for strict inequalities and a solid line for non-strict ones. Shade the appropriate region for each inequality, indicating where solutions exist. The feasible region is where all shaded areas overlap, representing all points that satisfy all inequalities in the system.
• What role do boundary lines play in solving systems of inequalities?
  • Boundary lines are essential as they determine the limits of each inequality within a system. Each boundary line separates the coordinate plane into two halves, one of which contains the solutions to that particular inequality. By identifying whether these lines are dashed or solid based on whether the inequality is strict or non-strict, we can accurately shade the feasible regions. Understanding these lines helps in visualizing how multiple constraints interact to define feasible solutions.
• Evaluate how systems of inequalities apply to real-world scenarios and linear programming.
  • Systems of inequalities are used extensively in real-world scenarios such as resource allocation, budgeting, and production scheduling, where multiple constraints must be satisfied. In linear programming, these systems help define feasible regions where potential solutions exist while maximizing or minimizing an objective function. For example, businesses might use systems of inequalities to determine how many products to produce within resource limits. The optimal solution can often be found at one of the vertices of the feasible region, showcasing how mathematical concepts translate into effective decision-making strategies.

Related terms

Feasible Region:

The area on a graph where all the inequalities in a system are satisfied, representing all possible solutions.

Linear Inequality:

An inequality that involves a linear expression and can be written in the form of $$ax + by \leq c$$, where $$a$$, $$b$$, and $$c$$ are constants.

Boundary Line:

The line that represents the equation of an inequality when converted into an equality; it helps to determine where the feasible region starts and ends.