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Chapter 2: Problem 44

Write an inequality that describes all points in the half-plane right of the \(y\) -axis.

Short Answer

Expert verified
The inequality is x > 0.

Step by step solution

01

Understanding the Half-Plane

The half-plane right of the y-axis includes all points with positive x-coordinates. This means any point (x, y) in this region satisfies x > 0.
02

Identifying the Proper Inequality Symbol

The inequality x > 0 is used to describe the set of points because we're looking for all points with x-values greater than 0, which lie to the right of the y-axis.
03

Writing the Inequality

We can now write the inequality that includes all such points. The correct inequality is simply x > 0, as it encompasses any point on the coordinate plane where x is greater than zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Plane
The coordinate plane is a two-dimensional surface on which we plot points, lines, and curves. It serves as a basic building block for understanding geometry and algebra concepts.
The plane is defined by two perpendicular axes: the horizontal x-axis and the vertical y-axis. Each point on this plane is represented as an ordered pair \(x, y\). This ordered pair indicates the exact position of the point relative to these axes.
Here鈥檚 what you should know about the coordinate plane:
  • It is divided by the x-axis and y-axis into four quadrants.
  • The point where the axes intersect is called the origin, denoted as \(0, 0\).
  • A positive x-coordinate means the point is to the right of the y-axis, and a positive y-coordinate means the point is above the x-axis.
Understanding the coordinate plane is essential for solving problems involving positions and movements on a flat surface. It helps visualize inequalities and other algebraic expressions.
X-coordinate
The x-coordinate is the first number in an ordered pair \(x, y\) that determines a point's horizontal position in the coordinate plane. It tells us how far left or right the point is from the origin.
Here are some key details about x-coordinates:
  • Positive values indicate positions to the right of the y-axis.
  • Negative values show positions to the left of the y-axis.
  • An x-coordinate of zero means the point lies directly on the y-axis.
In the context of inequalities, determining whether x is greater than, less than, or equal to a number can designate a specific region in the coordinate plane. For this exercise, focusing on an inequality like \(x > 0\) is critical, as it defines the region where all x-coordinates are positive, thus delineating the half-plane to the right of the y-axis.
Half-Plane
A half-plane refers to one side of a boundary line in the coordinate plane. When you draw a line, it splits the plane into two distinct regions, known as half-planes.
Here's what you should know about half-planes:
  • They are typically described using inequalities, such as \(x > 0\) or \(x < 0\).
  • One half-plane includes all points that satisfy the inequality.
  • The boundary line itself is not included unless the inequality is non-strict (e.g., \(x \geq 0\)).
The concept of the half-plane is vital for graphing solutions to inequalities and understanding which areas of the coordinate plane satisfy a given condition. In this exercise, the half-plane to the right of the y-axis is described by the inequality \(x > 0\). This includes every point with a positive x-coordinate, showcasing how we use inequalities to map regions on the plane.

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