Question

Describe the region in space defined by the inequalities. \(y \geq 3\)

Short answer

The region is the half-space above or on the plane \(y = 3\).

Step-by-step solution

Understand the Given Inequality

The inequality given is \(y \geq 3\). It means that you are looking at all the values of \(y\) that are greater than or equal to 3.

Visualize the Plane y = 3

Consider the equation \(y = 3\). This represents a horizontal plane in the three-dimensional space where the value of \(y\) is constant and equal to 3 for all values of \(x\) and \(z\).

Define the Half-Space

The inequality \(y \geq 3\) represents the region that includes all points above or on the plane \(y = 3\). This means that for any coordinates \((x, y, z)\), \(y\) must be greater than or equal to 3.

Interpret the Geometric Region

In three-dimensional space, this region is a half-space that extends infinitely in the positive \(y\) direction starting from the plane \(y = 3\). It covers all possible values of \(x\) and \(z\).

Key concepts

Half-Space

A half-space in three-dimensional space can be thought of as a portion of the space that is divided by a plane. When dealing with inequalities, such as the one specified in the exercise, we use these inequalities to define half-spaces. For the inequality \(y \geq 3\), the plane \(y = 3\) acts as a boundary. Everything on or "above" this plane, in terms of the y-axis direction, is included in what we consider the half-space.

In simpler terms, a half-space is an infinite region on one side of a dividing plane. For this example, any point with coordinates \((x, y, z)\), will be part of our half-space if \(y\) is 3 or greater. Here are some key points to remember about half-spaces:

• A half-space extends infinitely in three directions along whatever plane acts as the boundary.
• The inequality decides which side of the plane is included in the half-space.
• The boundary plane itself can be part of the half-space if the inequality allows it (such as \(\geq\) or \(\leq\)).

Three-Dimensional Coordinate System

The three-dimensional coordinate system is a framework used to specify any point in space using three numbers. These numbers usually represent the x, y, and z coordinates, allowing us to position any point uniquely in space. Much like a map, this system supports spatial understanding and navigation.

The three dimensions are usually imagined as axes on a graph:

• The x-axis typically runs left and right.
• The y-axis runs up and down.
• The z-axis runs forward and backward.

This coordinate system is essential when defining regions using inequalities, as seen in our exercise. For instance, the inequality \(y \geq 3\) restricts the minimum value of the y-coordinate while leaving the x and z coordinates unrestricted. This allows us to "carve out" a specific region of the three-dimensional space based on y's values.

Visualizing this coordinate system helps us understand and work with geometric shapes and regions effectively.

Geometric Visualization

Geometric visualization is an important skill that helps in understanding mathematical concepts by creating mental images of geometric scenarios. In this exercise, visualizing the inequality \(y \geq 3\) involves imagining a three-dimensional space with the plane \(y = 3\).

Picture this plane: it stretches infinitely across both the x and z axes, forming a kind of "ground" at the height of 3 on the y-axis. All points lying on this plane have a y-value of exactly 3. Now, consider that the inequality includes all points where y is more than or just equal to 3, generating a region above and on this plane. This region is infinite in size, spreading upwards in the "sky" above the plane.

Now, to aid your visualization:

• The plane divides the space into two sections: one where \(y < 3\) and the other where \(y \geq 3\).
• The half-space extends indefinitely upwards and doesn't constrain how the x or z values change.

Practicing geometric visualization can significantly enhance your ability to comprehend complex mathematical concepts by seeing them in a spatial form.