Problem 69 - Algebra and Trigonometry

Question

Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. $$-2\quad and\quad 5$$

Step-by-Step Solution

Answer

The distance between the numbers -2 and 5 on the number line is 7.

Step 1: Identifying the Numbers

The two numbers provided in this task are -2 and 5.

Step 2: Writing the Absolute Value Expression

The distance of two numbers on a number line can be found using the formula \(d = |a - b|\), where \(a∈R\) and \(b∈R\) are any two real numbers, and \(d\) represents the distance. So in this case, the absolute value expression to find the distance between -2 and 5 can be written as \(|5 - (-2)|\) or \(|-2 - 5|\). For the purposes of simplifying, we will go with the former i.e. \(|5 - (-2)|\).

Step 3: Evaluating the Absolute Value Expression

Solve the expression \(|5 - (-2)|\). Remember that subtracting a negative number is the same as adding a positive number, so this expression becomes \(|5 + 2|\), which simplifies to \(|7|\). The absolute value of 7 is indeed 7, because 7 is 7 units away from 0 on the number line.

Key Concepts

Distance on a Number Line

Understanding the concept of distance on a number line can be incredibly helpful when working with numbers. The number line is an imaginary line where each point represents a real number. It extends infinitely in both directions from zero. To find the distance between any two numbers on this line, you simply calculate the difference between them. However, the crucial part is that this difference is always taken as a positive value. This positive value is, in mathematical terms, called the "absolute value". For instance, the distance between

• -2
• 5

on a number line is calculated by subtracting one from the other and taking the absolute value to make it positive, because distance cannot be negative.So, their distance is expressed as

• \(|5 - (-2)| = |7| = 7\)

Thus, the distance on the number line is 7 units.

Evaluating Expressions

Evaluating expressions is a fundamental skill in algebra and mathematics as a whole. It involves simplifying an expression to find its value. When dealing with absolute values, we focus on ensuring that the outcome is always positive.Consider the expression

• \(|5 - (-2)|\)

To evaluate this, follow these simple steps:

• First, recognize that subtracting a negative number is the same as adding a positive.
• Transform the expression to \(|5 + 2|\).
• Simplify to get \(|7|\).
• Since absolute values strip away any negative sign (but 7 is positive), \(|7|\) is simply 7.

This is how evaluating expressions helps us find answers.

Real Numbers

Real numbers form the foundation on which most of arithmetic and algebra is based. They can be visualized on a number line, which shows that they include both rational numbers (like fractions and integers) and irrational numbers (like \(\sqrt{2}\) or \(\pi\)).Real numbers have some interesting properties:

• They are ordered: meaning you can compare them to state which is larger or smaller.
• They are dense: between any two real numbers, there's another real number.
• They form a complete set: which refers to the fact that every number can be accounted for on the number line.

In our exercise, the numbers -2 and 5 are real numbers, and they exist within this complete, ordered set. That’s why the concepts of distance and evaluating expressions of real numbers on a number line make sense, and why they work so beautifully in these calculations.