Problem 2 - Algebra 2 and Trigonometry

Question

Greg said that \(|a-b|=|b-a| .\) Do you agree with Greg? Explain why or why not.

Step-by-Step Solution

Answer

Yes, Greg is correct because \(|a-b| = |b-a|\) due to properties of absolute value.

Step 1: Understand the Absolute Value Property

The absolute value of any real number \(x\) is denoted by \(|x|\), and it represents the distance of \(x\) from 0 on the number line, regardless of the direction. Thus, it is always non-negative. In mathematical terms, \(|x| = x\) if \(x\geq 0\) and \(|x| = -x\) if \(x < 0\). This property indicates that absolute value disregards the sign of a number.

Step 2: Apply the Property to the Absolute Difference

The given expression is \(|a-b|=|b-a|\). By the definition of absolute value, we have:- \(|a-b|\) represents the distance between \(a\) and \(b\) on the number line.- \(|b-a|\) also represents the distance between \(b\) and \(a\), which is equivalent to \(|-(a-b)|\).Using the absolute value property, we know \(|-(a-b)|=|a-b|\). Therefore, \(|a-b|\) is always equal to \(|b-a|\).

Step 3: Conclude with Mathematical Equality

From the analysis, we see that changing the order of terms inside an absolute value operation does not change its outcome. Hence, \(|a-b| = |b-a|\) is a mathematical identity based on the properties of absolute values. Therefore, Greg's statement is indeed correct.

Key Concepts

Mathematical Identity

In mathematics, an identity is an equation that holds true for all values within its domain. The equation \(|a-b| = |b-a|\) is an example of a mathematical identity. This is because, no matter what real numbers you choose for \(a\) and \(b\), the expression remains valid.

This particular identity is rooted in the symmetric nature of absolute value. When evaluating absolute values, the sequence of numbers—whether \(a-b\) or \(b-a\)—is irrelevant because it calculates the distance between two points, not the direction. This concept can be utilized to simplify expressions and solve equations, knowing that reversing subtraction inside absolute value does not alter the result.

Distance on Number Line

Absolute value is fundamentally linked to the concept of distance on the number line. Consider the difference \(a-b\) where \(a\) and \(b\) are real numbers. The value \(|a-b|\) measures how far apart these two numbers are, without considering whether \(a\) is greater than \(b\) or vice versa.

Since distance is always non-directional, the expressions \(|a-b|\) and \(|b-a|\) depict the same measurement. They both reflect the true length between the points \(a\) and \(b\) on the number line, ensuring the measure is always a positive quantity or zero if \(a=b\).

• For example, if \(a = 7\) and \(b = 3\), then \(|a-b| = |7-3| = 4\) and \(|b-a| = |3-7| = 4\).

Non-negative Values

Another core principle of absolute values is that they yield non-negative results. Whether working with \(|a-b|\) or \(|b-a|\), the evaluated result is always zero or positive.

This arises from the definition of absolute value:

• \(|x| = x\) if \(x\) is non-negative.
• \(|x| = -x\) if \(x\) is negative, effectively converting \(x\) into a non-negative number.

With absolute values, any number's distance is portrayed positively, reinforcing the concept of non-negativity, vital for distances, which cannot inherently be negative.