Problem 49 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true for all real numbers \(a\) and \(b\). $$ |a-4|=|4-a| $$

Step-by-Step Solution

Answer

The statement \(|a-4| = |4-a|\) is true for all real numbers \(a\), as it holds true for both cases when \(a-4\) is non-negative and when \(a-4\) is negative.

Step 1: Understanding the absolute value expression

Absolute value is the distance of a number from 0 on the number line, it is always a non-negative number. The absolute value of an expression, e.g., \(|x|\), will be equal to the positive of the expression when the expression is positive or zero, and the negative of the expression when the expression is negative. In our case, we have two absolute value expressions \(|a-4|\) and \(|4-a|\). We will consider two cases

Step 2: Case 1: When \(a-4\) is non-negative

When \(a-4\) is non-negative (i.e., \(a-4 \ge 0\)), the left side of the equation becomes \(a-4\) because it's already positive or zero. The right side of the equation also simplifies to \(a-4\) because the negative of the expression inside the absolute value is \(4-a\), and since \(a-4 \ge 0\), we get: \(-(4-a) = a-4\). So in this case, both sides of the equation are the same, and the statement is true.

Step 3: Case 2: When \(a-4\) is negative

When \(a-4\) is negative (i.e., \(a-4 < 0\)), the left side of the equation becomes \(4-a\) because we need to take the negative of the expression inside the absolute value (\(4-a > 0\)). The right side of the equation remains as \(4-a\), because the expression inside the absolute value is already positive or zero (\(4-a \ge 0\)). So in this case, both sides of the equation are the same, and the statement is true.

Step 4: Conclusion

Since the statement holds true for both cases when \(a-4\) is non-negative and when \(a-4\) is negative, we can conclude that the statement \(|a-4| = |4-a|\) is true for all real numbers \(a\).

Key Concepts

Real Numbers

Real numbers are a fundamental part of mathematics and encompass all numbers that can be found on the number line. This includes both rational numbers, like integers and fractions, and irrational numbers, such as square roots of non-perfect squares and pi. Real numbers can either be positive, negative, or zero.

• Rational Numbers: These are numbers that can be expressed as fractions, where both the numerator and the denominator are integers (e.g., 3/4, -1, 7).
• Irrational Numbers: These cannot be expressed as simple fractions and include numbers like \(\sqrt{2}\) and π.

Understanding real numbers is crucial in solving mathematical problems since they form the backbone of most algebraic equations. Considering the nature of real numbers will help you analyze expressions correctly, as they ensure that the concepts applied, such as absolute values, cover every possible number on the line.

Mathematical Proof

A mathematical proof is a logical argument that demonstrates the truth of a given statement based on accepted mathematical techniques. Proofs are essential because they solidify the validity of mathematical statements. In proving the statement \(|a-4| = |4-a|\),

• We rely on properties of symmetry and the definition of absolute values.
• Using the definition of the absolute value, which accounts for both positive and negative states, directly shows equivalence between the two expressions.

To construct such a proof, we need to methodically apply rules and definitions to each part of the statement. In this particular case, we demonstrate the truth for both possible conditions of \(a-4\): when it is non-negative and when it is negative. This logical progression ensures that the proof is comprehensive and applies to all real numbers.

Case Analysis

Case analysis is a crucial method in solving mathematical statements that involve variables which behave differently under various conditions. By considering all potential scenarios, we form a complete understanding of the problem.

• In our problem, we have two cases based on whether \(a-4\) is non-negative or negative.
• Case 1 assumes \(a-4\) is non-negative, simplifying the equation by directly employing the property of absolute values on both sides.
• Case 2 assumes \(a-4\) is negative, adjusting the interpretation of the absolute values, which again renders both sides of the equation equal.

This methodical examination ensures every potential outcome is accounted for, confirming that the original statement holds true for any real number. Consequently, case analysis is an invaluable tool in both educational and professional mathematical environments.