Question

Use the given real number line to compute each distance. $$ \begin{array}{rrrrrrrrrrr} & A & & B & C & D & & E & & & \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline-4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \end{array} $$ $$ d(A, E) $$

Short answer

The distance between A and E is 7.

Step-by-step solution

Identify Coordinates of Points

Identify the coordinates of points A and E from the given number line. From the number line, the coordinate of point A is -4 and the coordinate of point E is 3.

Use the Distance Formula

Use the distance formula for two points on a number line: \[ d(A, E) = |E - A| \]Substitute the coordinates into the formula: \[ d(A, E) = |3 - (-4)| \]

Simplify the Expression

Simplify the expression inside the absolute value:\[ d(A, E) = |3 + 4| = 7 \]

Calculate the Distance

The absolute value of 7 is 7, so the distance between points A and E is 7.

Key concepts

absolute value

The concept of absolute value is essential for understanding how to find distances on a number line. Absolute value refers to the distance a number is from zero on the number line, regardless of direction. For instance, the absolute value of both -4 and 4 is 4. This is denoted by vertical bars, like this: \(|-4| = 4\).
In general, the absolute value of a number x is written as \(|x|\). Here are some points to remember:

• The absolute value of a positive number is the number itself. For example, \(|5| = 5\).
• The absolute value of a negative number is the positive version of that number. For example, \(|-3| = 3\).
• The absolute value of zero is 0. For example, \(|0| = 0\).

Knowing how to compute absolute values allows you to easily determine the distance between points on a number line. This is particularly useful for exercises like the one here, where you need to find the distance between points A and E.

distance formula

To find the distance between two points on a number line, we use a simple formula that involves absolute values. The distance formula states:
\[ d(A, B) = |B - A| \]
Here, \(d(A, B)\) represents the distance between points A and B, and \(A\) and \(B\) are their respective coordinates.
Let's quickly review the steps from the example exercise. The coordinates of \(A\) and \(E\) are -4 and 3, respectively. Substituting these into the distance formula gives us:
\[ d(A, E) = |3 - (-4)| \]
When we simplify the expression inside the absolute value, we get:
\[ d(A, E) = |3 + 4| = 7 \]
Finally, the distance between points \(A\) and \(E\) is 7 units. This formula works because absolute value ensures we're measuring the true distance, which is always positive.

number line

A number line is an essential tool for visualizing numbers, their relationships, and distances between them. It's a straight line where each point corresponds to a real number, spaced at uniform intervals. Understanding number lines simplifies many mathematical concepts, making them easier to grasp.
Here are some useful points about number lines:

• The center of the number line is typically 0.
• Positive numbers are located to the right of 0.
• Negative numbers are located to the left of 0.

In our example, we used a number line to identify the coordinates of points A and E. We then applied the distance formula to those coordinates to find the distance between them.
By visualizing these points on a number line, you can see the clear separation and confirm that the absolute value yields the true distance. Practicing with number lines will enhance your understanding of distances and magnitudes in math.