key term - Strict Inequalities

Definition

Strict inequalities are mathematical expressions that compare two values using the symbols '<' (less than) or '>' (greater than), indicating that one value is strictly less than or strictly greater than the other. These types of inequalities are commonly encountered in the context of systems of nonlinear equations and inequalities involving two variables.

5 Must Know Facts For Your Next Test

1. Strict inequalities are used to define the boundaries of a region in the coordinate plane, unlike non-strict inequalities that include the boundary points.
2. Solving a system of nonlinear equations and inequalities with two variables often requires the use of graphing, as the solution set may not be a simple linear region.
3. The solution set of a system of strict inequalities is typically a non-convex region, which means it may have curved or disconnected boundaries.
4. Strict inequalities are important in optimization problems, where the goal is to find the maximum or minimum value of a function subject to a set of constraints.
5. When solving a system of nonlinear equations and inequalities, the use of strict inequalities can help identify unique solutions or determine the feasibility of a given problem.

Review Questions

• Explain how strict inequalities are used to define the solution set of a system of nonlinear equations and inequalities with two variables.
  • Strict inequalities, such as $x < 3$ or $y > 2$, are used to define the boundaries of the solution set for a system of nonlinear equations and inequalities with two variables. Unlike non-strict inequalities that include the boundary points, strict inequalities exclude the boundary points, resulting in a non-convex region that may have curved or disconnected boundaries. The solution set is the intersection of the regions defined by the individual strict inequalities in the system.
• Describe the importance of strict inequalities in optimization problems involving systems of nonlinear equations and inequalities.
  • Strict inequalities play a crucial role in optimization problems, where the goal is to find the maximum or minimum value of a function subject to a set of constraints. The use of strict inequalities can help identify unique solutions or determine the feasibility of a given problem. For example, in a profit maximization problem, strict inequalities may be used to define the boundaries of the feasible region, which represents the set of all possible combinations of the decision variables that satisfy the given constraints. The optimal solution will then be found within this feasible region.
• Analyze how the use of strict inequalities in a system of nonlinear equations and inequalities with two variables can affect the characteristics of the solution set compared to the use of non-strict inequalities.
  • The use of strict inequalities in a system of nonlinear equations and inequalities with two variables can significantly affect the characteristics of the solution set compared to the use of non-strict inequalities. Strict inequalities exclude the boundary points, resulting in a non-convex region that may have curved or disconnected boundaries. This can lead to a more complex solution set that may not be a simple linear region, as would be the case with a system of linear equations and inequalities. The solution set may also have a different number of feasible solutions or even no feasible solutions, depending on the specific constraints defined by the strict inequalities. Understanding the impact of strict inequalities on the solution set is crucial when solving optimization problems or analyzing the behavior of nonlinear systems.

Related terms

Nonlinear Equations: Equations that cannot be expressed in the form of a linear equation, $ax + by = c$, where $a$, $b$, and $c$ are constants.

Nonlinear Inequalities: Inequalities that cannot be expressed in the form of a linear inequality, $ax + by \geq c$ or $ax + by \leq c$, where $a$, $b$, and $c$ are constants.

System of Equations and Inequalities: A set of two or more equations and/or inequalities that must be satisfied simultaneously.