Problem 151 - Prealgebra and Introductory Algebra

Question

Mathematics The distance \(d\) between point \(a\) and point \(b\) on the number line is given by the formula \(d=|a-b| .\) Find \(d\) when \(a=7\) and \(b=-12\)

Step-by-Step Solution

Answer

The distance \(d\) between the points \(a\) and \(b\) on the number line is \(19\).

Step 1: Substitute Values into the Formula

In the formula \(d = |a - b|\), substitute \(a\) with \(7\) and \(b\) with \(-12\). After substituting, the formula should be \(d = |7 - (-12)|\)

Step 2: Simplify the Expression Inside the Absolute Value

Simplify \(7 - (-12)\) to \(7 + 12\) which equals to \(19\).The formula becomes \(d = |19|\).

Step 3: Compute the Absolute Value

The absolute value of \(19\) is simply \(19\), as the absolute value of any positive number or zero is the number itself. So, \(d = 19\).

Key Concepts

Absolute Value

The concept of absolute value might initially seem like a fancy term, but it's quite simple once you break it down. Essentially, the absolute value of a number is its distance from zero on the number line without considering which direction from zero the number lies. So, whether a number is negative or positive, its absolute value is always a non-negative number.

Here's a simple way to remember this:

• If the number is positive or zero, its absolute value is the number itself. For example, the absolute value of 5 is 5.
• If the number is negative, the absolute value is the number without the negative sign. For example, the absolute value of -3 is 3.

In the context of distance, absolute value comes into play when calculating the "length" between two points. Regardless of direction, distance cannot be negative, hence the use of absolute value in its calculation.

Number Line

The number line is a fundamental visual tool used in mathematics to represent numbers in a straight, horizontal line, extending infinitely in both directions. Each point on the number line corresponds to a real number moving from left to right as you increase and from right to left as you decrease. The central point is usually zero, marking the division between positive and negative values.

A few key characteristics of a number line:

• Positive numbers are located to the right of zero.
• Negative numbers are located to the left of zero.
• The space between each pair of numbers is uniform and equal, representing a consistent scale.

When solving problems like finding the distance between two points, visualizing their positions on a number line can make it easier to understand the concept of their separation or distance from each other.

Distance Between Points

The distance between two points on a number line is a measure of how far apart they are. The standard formula to find this distance is surprisingly straightforward: the absolute value of the difference between the two numbers involved.

Here's how it works:

• Label the points as "Point A" and "Point B" with respective values, for example, \(a\) and \(b\).
• Calculate the difference by subtracting one point from the other, \(a - b\) or \(b - a\).
• Find the absolute value of that difference, \(|a - b|\), which represents the true distance, ensuring it's always non-negative.

To illustrate, let's look at the example from the exercise: to find the distance between 7 and -12, you would start with their difference, \(7 - (-12)\), simplifying to \(7 + 12 = 19\). The absolute value of 19 is simply 19, confirming the two points are 19 units apart on the number line.