Problem 54 - Algebra 1: Concepts and Skills

Question

Determine whether the statement is always, sometimes, or never true. Explain. The absolute value of a number is the same as the absolute value of the opposite number. In other words, \(|x|=|-x|\)

Step-by-Step Solution

Answer

The statement 'The absolute value of a number is the same as the absolute value of the opposite number' is always true. This is because the absolute value operation disregards the sign of the real number, thus the absolute values of any number and its opposite will indeed be the same.

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, disregarding the direction. So, whether you go to the right or to the left of zero, the distance is always a non-negative number, which is either positive or zero. For any real number x, the absolute value is denoted as \(|x|\).

Step 2: Applying Absolute Value to a Number

Consider a positive number 'p'. The absolute value of 'p', denoted \(|p|\), is simply 'p' because it's already positive. So, if \(x = p\), then \(|x|= p\).

Step 3: Applying Absolute Value to an Opposite Number

Now consider the number '-p', which is the opposite of 'p'. The absolute value of '-p', denoted \(-|p|\), will also be 'p' because the absolute value makes any real number non-negative. So, if \(x = -p\), then \(|x|= p\).

Step 4: Comparing the Absolute Values

From the previous observations, it's clear that for any real number 'x' and its opposite '-x', the absolute values are indeed the same, i.e., \(|x|=|-x|\). This happens because the absolute value of a number gives its magnitude without considering its sign.

Key Concepts

Understanding the Number Line

The number line is a fundamental concept in mathematics, representing all real numbers as points on a straight line. Imagine the line stretching indefinitely in both directions, with zero at the center. Numbers to the right of zero are positive, and those to the left are negative. The importance of the number line lies in its ability to visualize arithmetic operations and concepts such as the absolute value.

When dealing with absolute value, the number line becomes particularly helpful. For instance, if we take any number 'x' and its opposite '-x', we can see on the number line that they are the same distance from zero but in opposite directions. This visual representation on the number line underscores why the absolute values of 'x' and '-x' are equal, corresponding to the distance each point is from zero without regard to the direction.

What Are Real Numbers?

Real numbers encompass all the numbers we typically use, including both rational and irrational numbers. Rational numbers are those that can be expressed as the ratio of two integers, like \frac{1}{2} or 3. Irrational numbers, however, cannot be expressed as a simple fraction, examples being \(\sqrt{2}\) or \(\pi\). Both rational and irrational numbers are found on the number line.

When it comes to absolute value, it applies to all real numbers. Whether you're dealing with the square root of a non-perfect square, a negative fraction, or an irrational number, you can find the absolute value by determining the number's distance from zero on the number line.

Magnitude of a Number

The concept of the magnitude of a number is often associated with absolute value. Magnitude refers to the size or extent of a number, without considering its direction (positive or negative). In other words, it's the 'amount' of the number. For any real number, its magnitude is represented by its absolute value. So when we say the magnitude of '-5' or '5', we refer to the number '5', because we're describing how large the number is without the sign.

In our daily lives, magnitude is useful when the direction doesn't matter but the size does, such as when measuring the length of a piece of wood or the distance one has traveled.

Non-Negative Numbers

Non-negative numbers are exactly what they sound like: numbers that are not negative. This set includes all positive numbers and zero. They're located to the right of zero on the number line and include zero itself. These are particularly important in the context of absolute value, because the absolute value of any real number is always a non-negative number.

The concept is critical in areas such as accounting, where losses are represented by negative numbers and profits by non-negative numbers. Similarly, in mathematics, whenever we use the absolute value function, it ensures that the result is always in the realm of non-negative numbers, emphasizing that it is a measure of magnitude, not direction.