Question

To describe: The representation of equation \(y < 8\) in \({\mathbb{R}^3}\).

Short answer

The equation \(y < 8\) in \({\mathbb{R}^3}\) represents a half-space that consists of all points which are left to the plane \(y = 8\).

Step-by-step solution

Concept of \({\mathbb{R}^3}\)

\({\mathbb{R}^3}\)is the three dimensional coordinate system which contains\(x\),\(y\), and\(z\)-coordinates.

Represent the given equation in \({\mathbb{R}^3}\)

The equation \(y < 8\) in \({\mathbb{R}^3}\) represents the set \(\{ (x,y,z)\mid y < 8\} \), which is the set of all points in \({\mathbb{R}^3}\) whose \(y\) coordinate is less than \(8\) and \(x\), \(z\)-coordinates are any values.

The equation \(y = 8\) in \({\mathbb{R}^3}\) is sketched as shown in Figure \(1\).

Thus, the equation \(y < 8\) in \({\mathbb{R}^3}\) represents a half-space that consists of all points which are left to the plane \(y = 8\).