Question

Determine the location of a point \((x, y, z)\) that satisfies the condition(s). \(|y| \leq 3\)

Short answer

The point \((x, y, z)\) that satisfies the condition \(|y| \leq 3\) exists in a plane where x and z can be any real number and y lies between -3 and 3, inclusive. This forms an infinite slab in R3.

Step-by-step solution

Understanding Absolute Value Equation

The absolute value of a number is its distance from zero. Thus, when we have \(|y| \leq 3\), this means that the value of y in \((x, y, z)\) could be any number that has a distance from zero that is less than or equal to 3. This could be any number between -3 to 3, inclusive.

Describing the Potential Values of y

Since \(|y| \leq 3\), the possible values of y will be from -3 to 3. Hence the y coordinate in the point \((x, y, z)\) could be any number between -3 and 3, inclusive, to satisfy the given condition. The values of x and z can be any real number, they are not constrained by the given condition.

Formulating the Answer

To formulate the space in which \((x, y, z)\) resides if it satisfies the given condition, all real numbers for x and z and the y-coordinate that ranges from -3 to 3 are included.