Problem 74 - Algebra and Trigonometry

Question

Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. $$-5.4\quad and\quad -1.2$$

Step-by-Step Solution

Answer

The distance between -5.4 and -1.2 is 4.2.

Step 1: Write the Absolute Value Expression

The problem asks for the distance between \(-5.4\) and \(-1.2\), which can be expressed as the absolute value of their difference: \( |-5.4 - (-1.2)| \). Notice that subtracting a negative number is the same as adding a positive number, hence the expression can be simplified to: \( |-5.4 + 1.2| \).

Step 2: Evaluate the Absolute Value

Simplify the expression inside the absolute value symbol: \(-5.4 + 1.2 = -4.2\). The absolute value of \(-4.2\) is its distance from 0 on the number line, which is \(4.2\). Hence, \( |-4.2| = 4.2 \).

Key Concepts

Distance Between Numbers

Understanding the distance between numbers is crucial when learning about absolute value. It's similar to measuring the length of a segment between two points on a number line. Regardless of the direction, the distance is always a positive quantity or zero.

In the context of the exercise, finding the distance between (-5.4) and (-1.2) requires you to consider their positions on the number line. Because both numbers are negative, they are located to the left of zero. To determine the distance or the absolute difference, you subtract the smaller number from the larger one (considering their numerical value without regard to sign) or vice versa, as the order of subtraction does not affect the distance.

This operation results in an expression that when solved, gives you the absolute value—a non-negative result representing the distance. Irrespective of which number is subtracted from the other, the absolute value ensures a positive distance, eliminating any confusion due to the negative signs.

Evaluating Expressions

Evaluating expressions, particularly those involving absolute values, is a fundamental skill in algebra. An expression is a combination of numbers, variables (placeholders for numbers), and operations such as addition and subtraction. The act of evaluating simplifies the expression to find a single numerical value.

For the given problem, evaluating the expression involves two main steps. Firstly, you perform the operations inside the absolute value signs. In this case, we start by simplifying the expression (-5.4 + 1.2), treating it just like any numerical expression without the absolute value. Then, we subtract (-1.2) from (-5.4) mindful that subtracting a negative is the same as adding the opposite.

Once you've simplified the inner expression, you then evaluate the absolute value, which converts any negative result to positive, thus finding the actual distance. The process of evaluation is methodical and allows you to decipher complex expressions step by step.

Negative Numbers

Negative numbers often represent a challenge, especially for those just beginning to explore more complex aspects of mathematics. These numbers are values that are less than zero, typically represented with a minus sign in front of them.

In the given exercise, you deal with negative numbers when looking to express distance. It's important to understand that while the original values (-5.4) and (-1.2) are negative, indicating their position to the left of zero on the number line, the distance measured as the absolute value is a positive value or zero. This concept reflects the idea that distance itself cannot be negative.

Furthermore, when negative numbers are involved in calculations, remember that two negatives make a positive ((-1) * (-1) = 1). This rule applies to subtraction as well, as subtracting a negative number is equivalent to adding its positive counterpart. Mastering how to work with negative numbers is essential for correctly evaluating expressions and understanding the absolute value as a measure of distance.