Problem 66 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the properties of the absolute value to show that each of the following is true. (a) \(|a-b| \leq|a|+|b|\) (b) \(|a-b| \geq|a|-|b|\) (c) \(|a+b+c| \leq|a|+|b|+|c|\)

Step-by-Step Solution

Answer

The inequalities are proven using triangle and reverse triangle inequalities.

Step 1: Understanding the Triangle Inequality for Part (a)

To solve **(a) \(|a-b| \leq |a| + |b|\)**, we need to use the triangle inequality property, which states that for any real numbers \(a\) and \(b\), the absolute value of their sum is less than or equal to the sum of their absolute values: \(|x + y| \leq |x| + |y|\). By substituting \(x = a\) and \(y = (-b)\), we have \(|a + (-b)| \leq |a| + |-b|\). Since \(|-b| = |b|\), we get \(|a-b| \leq |a| + |b|\). This shows Part (a), using the property that the absolute value of \((-b)\) is \(b\).

Step 2: Understanding the Reverse Triangle Inequality for Part (b)

For **(b) \(|a-b| \geq |a| - |b|\)**, we use the reverse triangle inequality, which states: \(|x - y| \geq ||x| - |y||\). We can take \(x = a\) and \(y = b\), such that \(|a - b| \geq ||a| - |b||\). By the definition of absolute value, \(||a| - |b|| = |a| - |b|\) if \(|a| \geq |b|\) and equals \(|b| - |a|\) otherwise, but the former condition is covered since we have \(|a-b| \geq |a| - |b|\). Thus, this proves Part (b).

Step 3: Extending to Three Variables for Part (c)

For **(c) \(|a+b+c| \leq |a| + |b| + |c|\)**, we extend the triangle inequality to three terms. Firstly, consider \(x = a + b\) and \(y = c\), then apply the triangle inequality \(|x + y| \leq |x| + |y|\), resulting in \(|a + b + c| \leq |a + b| + |c|\). By the triangle inequality again, we have \(|a + b| \leq |a| + |b|\). Therefore, \(|a + b + c| \leq |a| + |b| + |c|\), which completes the proof for Part (c).

Key Concepts

Absolute Value Properties

Absolute value is a measure of the magnitude of a number without considering its direction on the real number line. It is denoted by vertical bars (e.g., \(|a|\)). When looking at properties of absolute value, some key rules come into play that make it easier to analyze mathematical expressions.

Here are some fundamental properties:

• \(|a| \geq 0\): The absolute value of any real number is always non-negative.
• If \(|a| = 0\), then \(a = 0\).
• \(|ab| = |a||b|\): The absolute value of a product is the product of the absolute values.
• \(|a/b| = |a|/|b|\), where \(b eq 0\): Similarly, the absolute value of a quotient is the quotient of the absolute values.
• \(|-a| = |a|\): The absolute value of a number is the same as the absolute value of its negative.

Understanding these properties is essential when solving problems involving inequalities and variables. They form the backbone for more complex concepts, like the triangle inequality, used to compare sums and differences of numbers.

Reverse Triangle Inequality

The reverse triangle inequality is a complementary concept to the triangle inequality, offering another way to compare magnitudes of differences of numbers. It states:

• \(|a - b| \geq ||a| - |b||\)

This inequality helps us understand how the difference between two numbers relates to the absolute difference of their magnitudes.

When using this property, there's an important consideration:

• If \(a\) and \(b\) are real numbers such that \(|a| \geq |b|\), then \(|a - b| \geq |a| - |b|\).

This is crucial when determining lower bounds in problems involving absolute value differences. It provides a more accurate understanding of comparisons between the magnitudes of differences.

Extension to Multiple Variables

In problems involving more than two variables, the concept of triangle inequality can be extended to handle multiple terms. A classic example involves three variables, like the expression \(|a+b+c|\).

This extension uses the principle that the absolute value of a sum involving more than two terms is less than or equal to the sum of the absolute values of each term:

• \(|a + b + c| \leq |a| + |b| + |c|\)

To prove this, you can apply the two-variable triangle inequality multiple times:

• First, consider \(a + b\) as a single entity: \(|a + b| \leq |a| + |b|\)
• Then, add the third term: \(|a + b + c| \leq |a + b| + |c|\) which simplifies further to \(|a| + |b| + |c|\)

By systematically breaking down the sum, it is easier to visualize and understand how the inequality holds for more complex expressions.