Problem 150 - 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 \(a=6\) and \(b=-15\)

Step-by-Step Solution

Answer

The distance \(d\) is 21.

Step 1: Substituting values

First, substitute the given values into the equation \(d = |a - b|\). Therefore, the equation becomes \(d = |6 - (-15)|\).

Step 2: Solving the equation

Next, simplify inside the absolute value brackets. Two minus signs right next to each other equals a plus: \(d = |6 + 15|\). This simplifies to \(d = |21|\).

Step 3: Getting the distance

The absolute value of 21 is 21. Therefore, the distance between point a and point b \(d = 21\).

Key Concepts

Number Line

Picture a straight horizontal line with numbers placed at equal intervals along it. This is what we call a number line. It gives us a visual representation of numbers where positive values are to the right and negative values are to the left from a neutral zero point in the center.

Using a number line can be extremely helpful for understanding concepts like distance between two points. The purpose is to show the positions of numbers in an ordered way, and to visually represent operations on numbers, such as addition and subtraction. In the context of our exercise, it helped to visualize the distance between the points when given their numerical locations, making it clearer how far apart they are.

Absolute Value

The absolute value of a number is a measure of how far the number is from zero on the number line, without considering which direction from zero the number lies. It is always a non-negative number.

Mathematically, the absolute value of a number a is denoted as |a|. To simplify, think of it like this: if a is positive, |a| is just a; if a is negative, |a| is -a, because we flip the sign to get a distance, which is always positive. For example, the absolute value of both 7 and -7 is 7. In our exercise, we used the absolute value to determine the distance between two points, since distance is inherently a positive measurement.

Substituting Values

When given a formula or an equation, you'll often need to replace variables with specific numbers. This process is called substituting values.

It's like following a recipe. Just as you might substitute olive oil for vegetable oil in a recipe, you substitute numerical values for variables in an equation. To ensure accuracy, it's crucial to replace the variables with numbers carefully, respecting the mathematical operations. In our exercise, substituting values accurately was the first step towards finding the distance between the two points on the number line.

Solving Equations

An equation is like a balance scale. When we solve equations, we want to find the value for the unknown that makes the equation true, maintaining that perfect balance.

We follow specific steps, like combining like terms, isolating the variable, and performing the same operations on both sides of the equation. Patience and attention to detail can make complex equations solvable. In our exercise, after substituting the values into the distance equation, we simplified within the absolute value and solved it to find that the distance between two points was 21 units.