Problem 57 - Calculus: One and Several Variables

Question

Knowing that \(|a+b| \leq|a|+|b| \quad\) for all real \(a . b\) show that $$|a|-|b| \leq|a-b| \quad \text { for all real } a . b$$

Step-by-Step Solution

Answer

The inequality \(|a|-|b| \leq |a-b|\) is true for all real numbers \(a\) and \(b\).

Step 1: Integrating the given inequality

Start with the given inequality \(|a+b| \leq |a|+|b| \). Substitute \(b\) with \(-b\) to get \(|a-b| \leq |a|+|-b| \).

Step 2: Simplifying equation

We know for all real numbers the absolute value of a number is either positive or zero, so \(|-b|\) is equal to \(|b|\). Therefore, \(|a-b| \leq |a|+|b| \).

Step 3: Deriving target inequality

Rearrange the equation from step 2 by subtracting \(|b|\) from both sides. It leads us to the desired inequality, which is \(|a|-|b| \leq |a-b|\).

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics. It simply tells us how far a number is from zero on the number line. For example, the absolute value of either 5 or -5 is 5. The notation for absolute value uses vertical bars, like this:

• For any number \(x\), the absolute value is indicated by \(|x|\).
• It is defined as \(|x| = x\) if \(x\) is positive or zero, and \(|x| = -x\) if \(x\) is negative.

The absolute value tells us about magnitude without considering direction or sign. In the inequality \(|a+b| \leq |a|+|b|\), we are seeing that combining two numbers and taking their absolute value never exceeds the sum of their absolute values. This inequality is known as the **Triangle Inequality**. It reflects a general rule about distances in a geometric sense, captured here with absolute values.

Real Numbers

Real numbers include all the numbers on the continuous number line. These extend from negative infinity to positive infinity and consist of rational and irrational numbers. Real numbers are crucial in mathematics because they can represent quantities with complete precision.

• **Rational Numbers:** These are numbers that can be expressed as the fraction of two integers, such as \(\frac{2}{3}\) or \(-4\).
• **Irrational Numbers:** These cannot be written as a simple fraction, examples include \(\sqrt{2}\) or \(\pi\).

The concept of real numbers encompasses fractions, whole numbers, and integers, forming a broad set that is used in many areas of mathematics. When working with inequalities like \(|a|-|b| \leq |a-b|\), we are operating within the scope of real numbers. This means we assume there is a value for every possible amount between any two numbers. This property is what allows us to perform algebraic manipulations to establish the inequality for any real numbers \(a\) and \(b\).

Inequality Proofs

Inequality proofs such as \(|a|-|b| \leq |a-b|\) show us relationships between entities where one is generally less than or equal to another. Proving inequalities in mathematics is often about demonstrating the truth of these relationships through logical reasoning and manipulation of algebraic expressions.

• **Step-by-Step Manipulation:** In the example problem, we started from a known inequality \(|a+b| \leq |a|+|b|\) and manipulated it to derive the target inequality.
• **Substitution and Simplification:** By substituting \(b\) with \(-b\), we used the properties of absolute values to simplify the inequality. This step is crucial as it paves the way for rearranging terms correctly.
• **Rearranging Terms:** This often involves adding or subtracting the same expression from both sides. In our example, subtracting \(|b|\) from both sides yielded the required inequality.

Mastering inequalities involves a good grasp of basic algebra and logical reasoning, allowing one to ascertain the relationships between expressions under specific conditions. It's essential to carefully perform each step to ensure the inequality is preserved throughout the process.