Problem 33 - College Algebra

Question

express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. \(-2\) and 5

Step-by-Step Solution

Answer

The distance between -2 and 5 is 7.

Step 1: Understanding absolute value

The absolute value of a number is its distance from zero. This means that it equals the number itself if the number is positive or zero, and it equals the negative of the number if the number is negative. We denote the absolute value of a number \(a\) by \(|a|\).

Step 2: Setting up the absolute value formula

To find the distance between the numbers -2 and 5, we set up an absolute value. Expression for them would be \(|-2 - 5|\).

Step 3: Compute the absolute value

We first simplify inside the absolute value: \(|-2 - 5| = |-7|\). The absolute value of -7 is 7 because absolute value makes any negative number positive. So, the distance between -2 and 5 is 7.

Key Concepts

Absolute Value Expression

When we talk about absolute value expressions, we're referring to a mathematical notation that captures distance from zero on the number line, without considering direction. Think of it as the 'numerical distance' without any negative signs.

For example, if you have the expression \( |x| \), this reads as 'the absolute value of x'. This is a way of asking 'how far is x from zero?'. It's a useful tool particularly in geometry, physics, and engineering where actual distances or magnitudes are necessary, regardless if the direction is positive or negative.

An absolute value expression can also include more complex operations inside the bars, such as \( |3x - 2| \), which would mean 'the absolute value of the expression 3x minus 2'. In such cases, you always perform the operations inside first before applying the concept of absolute value.

Evaluating Absolute Value

The process of evaluating absolute value requires two primary steps. First, simplify the expression inside the absolute value bars as if you were solving any other algebraic expression. Second, apply the absolute value operation itself.

Let's take a closer look using the expression from our initial exercise \( |-2 - 5| \). The first step is to combine the numbers inside, giving us \( |-7| \). As the second step, we evaluate the absolute value of -7. Since absolute value represents distance, which is always positive or zero, \( |-7| = 7 \). It doesn't matter whether the number inside is positive or negative - absolute value strips away the sign. In simpler terms, when you evaluate absolute value, you're always seeking a non-negative result.

Remember, the rule is simple: if the number is positive or zero, it stays the same; if it's negative, it becomes positive. This is why \( |-7| \), where -7 is negative, results in 7, a positive number.

Absolute Value of a Number

The absolute value of a number is a basic concept but crucial in understanding how we measure distances and magnitudes in mathematics. It pertains to the non-negative value of a number on the number line, regardless of its original sign. So, the absolute value of both 3 and -3 is 3 because both points are 3 units away from zero on the number line.

In mathematical terms, for any real number 'a', the absolute value is defined by two conditions: \( |a| = a \) if 'a' is greater than or equal to zero, and \( |a| = -a \) if 'a' is less than zero. These conditions ensure that the output is always a positive number or zero.

Understanding that the absolute value is the measure of distance on the number line is essential. This concept is widely used in various branches of mathematics, including calculus and statistics, and is fundamental in understanding the behavior of functions and solving equations involving absolute values.