≥

Definition

The symbol '≥' represents the concept of 'greater than or equal to' in mathematics, establishing a relationship between two values. This symbol is crucial in expressing linear inequalities, where one side of the inequality can either exceed or be equal to the other side. It helps define boundaries in mathematical expressions and is foundational in various applications like optimization and systems of inequalities.

5 Must Know Facts For Your Next Test

1. '≥' can be used in inequalities to indicate that a variable can take on values that are either greater than or exactly equal to a specified value.
2. In graphical representations, the boundary line for '≥' is solid, indicating that points on the line are included in the solution set.
3. '≥' allows for multiple solutions, as it encompasses all values above and including the point specified by the inequality.
4. Inequalities involving '≥' can be solved using similar methods as equations, such as isolating variables and testing intervals.
5. In linear programming, '≥' constraints help define the feasible region where optimal solutions can be found for maximizing or minimizing objective functions.

Review Questions

• How does the symbol '≥' change the interpretation of a linear inequality compared to just using '>'?
  • '≥' includes all values that are greater than the specified value, as well as the value itself. This means that when you have an inequality like x ≥ 5, both x = 5 and any value greater than 5 are part of the solution set. In contrast, with '>', the solution would exclude the value itself (x > 5 would not include 5). This distinction is important when graphing inequalities, as it affects whether points on the boundary line are included.
• Explain how '≥' is represented graphically and its significance in determining feasible solutions within a system of inequalities.
  • '≥' is represented by a solid boundary line on a graph because it indicates that points on the line are included in the solution set. The area above this line represents all values that satisfy the inequality. In a system of inequalities, identifying these regions helps to determine where feasible solutions lie. The intersection of multiple regions defined by different inequalities creates the feasible region essential for solving linear programming problems.
• Evaluate the impact of using '≥' in linear programming problems when determining optimal solutions for an objective function.
  • Using '≥' in linear programming defines constraints that limit possible solutions while still allowing for flexibility within certain bounds. This can significantly affect the feasible region, shaping it to include areas where values meet or exceed specific requirements. As these constraints are integrated into an objective function aimed at maximization or minimization, they help identify optimal solutions within those bounded areas. Thus, understanding how '≥' influences constraints is key to successful optimization strategies.

Related terms

Linear Inequality: An inequality that involves a linear expression and can be written in the form ax + b > c, ax + b < c, ax + b ≥ c, or ax + b ≤ c.

Boundary Line: A line that represents the solutions of an equation or inequality in a coordinate system, separating different regions where different conditions apply.

Feasible Region: The set of all possible points that satisfy a given set of inequalities, representing potential solutions for optimization problems.