Half-Plane

Definition

A half-plane is a region of the coordinate plane that is divided by a line. It represents all the points on one side of a given line, including the line itself. The concept of half-planes is particularly important in the context of graphing linear inequalities.

5 Must Know Facts For Your Next Test

1. The half-plane represents all the points on one side of a given line, including the line itself.
2. The boundary line of a half-plane is the line that divides the plane into two regions and represents the points that satisfy the equality in a linear inequality.
3. The direction of the inequality (greater than or less than) determines which half-plane is the solution set for the inequality.
4. Shading is used to visually represent the half-plane that satisfies a given linear inequality on a graph.
5. The concept of half-planes is crucial in understanding the graphical representation of linear inequalities, which is an important topic in elementary algebra.

Review Questions

• Explain how the direction of a linear inequality (greater than or less than) determines the half-plane that represents the solution set.
  • The direction of the inequality (greater than or less than) determines which half-plane is the solution set for the inequality. If the inequality is $ax + by \geq c$, the half-plane that includes the points satisfying the inequality is the half-plane that contains the points on the same side of the boundary line as the point $(0, c/b)$. Conversely, if the inequality is $ax + by > c$, the half-plane that includes the points satisfying the inequality is the half-plane that does not contain the point $(0, c/b)$.
• Describe the relationship between the boundary line of a half-plane and the points that satisfy the equality in a linear inequality.
  • The boundary line of a half-plane represents the points that satisfy the equality in a linear inequality. In other words, the boundary line is the set of points that make the inequality an equality. For example, if the inequality is $2x + 3y \geq 12$, the boundary line is the line $2x + 3y = 12$, and the half-plane that satisfies the inequality is the region that includes the points on the same side of the line as the point $(0, 4)$.
• Explain how the concept of half-planes is used to graph linear inequalities and discuss the importance of this concept in elementary algebra.
  • The concept of half-planes is crucial in understanding the graphical representation of linear inequalities, which is an important topic in elementary algebra. By recognizing that a linear inequality divides the coordinate plane into two half-planes, one can easily graph the solution set by shading the appropriate region. This visual representation helps students understand the relationship between the inequality, the boundary line, and the solution set. The ability to graph linear inequalities is a fundamental skill that is built upon in more advanced algebra and geometry concepts, making the understanding of half-planes essential for success in elementary algebra.

Related terms

Linear Inequality: A linear inequality is an inequality that can be written in the form $ax + by \geq c$ or $ax + by > c$, where $a$, $b$, and $c$ are real numbers.

Boundary Line: The boundary line of a half-plane is the line that divides the plane into two regions. It represents the points that satisfy the equality in a linear inequality.

Shading: Shading is the process of highlighting the region of the coordinate plane that satisfies a given linear inequality, which corresponds to the half-plane.