Problem 23 - Essential Calculus Early Transcendentals

Question

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(y<8\)

Step-by-Step Solution

Answer

The region is the space below the plane \(y = 8\) in \(\mathbb{R}^3\).

Step 1: Understand the Inequality

The inequality given is in the form of a linear inequality in three-dimensional space. Specifically, this inequality only involves the variable \(y\), suggesting a condition solely related to the \(y\)-coordinate.

Step 2: Fixed Condition on the y-direction

The inequality \(y < 8\) describes a condition that is dependent on the \(y\)-coordinate of any point in the space. This implies any point that satisfies this inequality must have a \(y\)-value less than 8.

Step 3: Imagining the Region in 3D Space

In three-dimensional space, \(y < 8\) corresponds to the set of points that lie below the horizontal plane defined by \(y = 8\). This horizontal plane is parallel to the \(xz\)-plane and is where the \(y\)-coordinate equals 8.

Step 4: Conclusion of the Region

Thus, the region represented by \(y < 8\) is the entire space in \(\mathbb{R}^3\) below the plane \(y = 8\), extending infinitely in the \(x\)- and \(z\)- directions and all directions below \(y = 8\).

Key Concepts

Linear Inequality

A linear inequality compares two expressions and describes a region of a coordinate system where a certain condition holds true. It is like an equation, but instead of an equals sign, you will see inequality symbols such as \(<\), \(\leq\), \(>\), or \(\geq\). By using linear inequalities, we can define parts of three-dimensional space, which are relevant in many mathematical and real-world applications. Linear inequalities can help in pinpointing where certain conditions are met, ensuring clarity in problem-solving. In the given exercise, the inequality \(y < 8\) defines a specific volume of space where the \(y\)-values of all points must be less than 8. This fundamental concept helps determine which parts of the space comply with the stated conditions. When working with inequalities in three-dimensional contexts, it's essential to visualize them to grasp how the inequality changes the feasible region.

Coordinate System

The coordinate system is a way to specify the position of points in space using coordinates, which act like a set of instructions. In three dimensions, we have the \(x\), \(y\), and \(z\) coordinates. These axes intersect perpendicularly at a common point known as the origin. They are fundamental to defining locations and geometrical figures in space.
An intuitive understanding of this system involves imagining the system as a set of reference lines that aid in pinpointing the exact location of any point. In our problem, the inequality \(y < 8\) specifically refers to the \(y\)-coordinate aspect. It tells us that for any point residing in the allowed region, its \(y\)-coordinate should be less than 8. In essence, the coordinate system provides the framework to interpret and apply geometric principles within a 3D context.

3D Geometry

Three-dimensional geometry studies the properties of shapes and figures that have depth in addition to height and width. It examines how these entities occupy space and how they interrelate. In 3D geometry, "planes" are flat, two-dimensional surfaces extending across the third dimension. In our case, the plane \(y = 8\) exists parallel to the \(xz\)-plane, cutting across the \(y\)-axis at 8.
Understanding this concept allows us to picture the region defined by \(y < 8\).

• Everything below the plane \(y = 8\) is included.
• It extends indefinitely in the downward \(y\) direction.
• The region has no bounds in the \(x\) and \(z\) directions.

This gives us a visual idea of where points satisfying the inequality can exist in 3D space, ensuring the "3D geometry" part of the exercise is clear.