Question

The distance between \(z\) and 0 is less than 8 .\(y\) is at least six units from \(0 .\)

Short answer

-86

Step-by-step solution

Understanding the Distances

When it's said that the distance from \(z\) to 0 is less than 8 units, it means \(z\) can be anywhere on the number line from -7 to 7 (excluding -8 and 8 as the distance is strictly less than 8). That is, -8<z<8.

Determining Range for y

Then, consider the statement that \(y\) is at least six units from 0. This means \(y\) can be positioned six units in either direction on the number line. Therefore, \(y\) can be less than -6 or greater than 6. That is, y <-6 or y>6.

Key concepts

Number Line

The number line is a visual representation that helps us understand numbers and their relationships. Imagine a line where numbers are positioned according to their value. Zero is usually placed at the center, with positive numbers extending to the right and negative numbers extending to the left. This organization makes it easy to visualize addition, subtraction, and the positions of different numbers with respect to each other.
For instance, when considering complex numbers like in the exercise, a number line can simplify the concept of distance from a specific point. By marking distances from the point zero, we can spot where other numbers lie and how far they are from this central point.

Distance

Distance on a number line refers to how far apart two points are. It is a measure of the length of the segment between those points.
In mathematical terms, the distance between a number, say \( z \), and zero can be expressed as \( |z| \). This represents the absolute value of \( z \), which signifies its distance from zero without considering its direction.
In the exercise, the distance condition was \( |z| < 8 \). This implies that \( z \) can be any number within 8 units from zero, either to the left or right. Thus, \( z \) can range between -7 and 7, as it can be at most 7 units away, but not exactly 8 units, being strictly less than 8.

Inequalities

Inequalities are mathematical statements that compare two values showing if one is smaller, larger, or equal relative to the other. These are represented by symbols such as <, >, ≤, and ≥.
In the exercise, the inequality \(-8 < z < 8\) means that \( z \) must be greater than -8 and less than 8. It describes a range of possible values for \( z \). This is useful for understanding not just fixed points but intervals, offering a flexibility in positioning.
Similarly, the condition \( y < -6 \) or \( y > 6 \) shows that \( y \) must be at a distance of at least 6 units away from zero. Therefore, \( y \) can either lie in the region beyond -6 to the left or beyond 6 to the right, emphasizing how inequalities capture ranges and constraints beautifully.