Inequality Symbols

Definition

Inequality symbols are mathematical symbols used to compare the relative size or magnitude of two values. These symbols indicate whether one quantity is greater than, less than, or equal to another quantity, allowing for the representation of inequalities in mathematical expressions and equations.

5 Must Know Facts For Your Next Test

1. Inequality symbols are essential in the context of graphing linear inequalities, as they determine the direction and shading of the solution region on the coordinate plane.
2. The solution set for a linear inequality is the set of all points that satisfy the inequality, and this set is represented by a half-plane in the coordinate plane.
3. The type of inequality symbol used (>, <, ≥, or ≤) determines the orientation of the half-plane, which can be either above or below the boundary line.
4. Solid inequality symbols (≥ and ≤) indicate that the boundary line is included in the solution set, while open inequality symbols (> and <) indicate that the boundary line is not included.
5. Correctly interpreting and applying inequality symbols is crucial for accurately graphing and solving linear inequalities in the coordinate plane.

Review Questions

• Explain how the type of inequality symbol used (>, <, ≥, or ≤) affects the orientation and shading of the solution region when graphing a linear inequality.
  • The type of inequality symbol used determines the orientation and shading of the solution region when graphing a linear inequality. For example, if the inequality is $y \geq 2x + 1$, the solution region will be the half-plane above the line $y = 2x + 1$, including the boundary line itself. In contrast, if the inequality is $y > 2x + 1$, the solution region will be the half-plane above the line $y = 2x + 1$, but the boundary line will not be included in the solution set. The solid inequality symbol (≥) indicates that the boundary line is part of the solution, while the open inequality symbol (>) excludes the boundary line.
• Describe the differences between the solution sets of the linear inequalities $x \leq 3$ and $x < 3$, and explain how the inequality symbols affect the inclusion or exclusion of the boundary point.
  • The linear inequality $x \leq 3$ has a solution set that includes all values of $x$ that are less than or equal to 3, including the boundary point $x = 3$. In contrast, the linear inequality $x < 3$ has a solution set that includes all values of $x$ that are strictly less than 3, excluding the boundary point $x = 3$. The solid inequality symbol (≤) indicates that the boundary point is included in the solution set, while the open inequality symbol (<) excludes the boundary point. This difference in the inclusion or exclusion of the boundary point can be crucial when solving problems involving linear inequalities and graphing their solution regions on the coordinate plane.
• Analyze the impact of using different inequality symbols (>, <, ≥, or ≤) on the solution set of a system of linear inequalities, and explain how this affects the graphical representation of the feasible region.
  • The choice of inequality symbols in a system of linear inequalities can significantly impact the solution set and the graphical representation of the feasible region. For example, if a system of linear inequalities includes the inequality $x + y \leq 4$, the feasible region will be the area on the coordinate plane that satisfies this inequality, which is the half-plane below the line $x + y = 4$, including the boundary line. However, if the inequality is changed to $x + y < 4$, the feasible region will be the area strictly below the line $x + y = 4$, excluding the boundary line. This change in the inequality symbol can alter the shape and size of the feasible region, which is crucial when solving optimization problems or making decisions based on the constraints represented by the system of linear inequalities.

Related terms

Greater Than (>): The symbol used to indicate that one value is larger or greater than another value.

Less Than (<): The symbol used to indicate that one value is smaller or less than another value.

Greater Than or Equal To (≥): The symbol used to indicate that one value is greater than or equal to another value.