Problem 33 - The Calculus with Analytic Geometry

Question

Prove that if \(a\) and \(b\) are any numbers, then \(|a-b| \leq|a|+|b|\). (HINT: Write \(a-b\) as \(a+(-b)\) and use Theorem 1.2.8.)

Step-by-Step Solution

Answer

By rewriting a - b as a + (-b) and applying Theorem 1.2.8, we get |a - b| ≤ |a| + |b|.

Step 1: Rewrite the expression

Rewrite the expression |a - b| as |a + (-b)|. This helps in utilizing properties involving addition.

Step 2: Apply Theorem 1.2.8

Theorem 1.2.8 states that for any real numbers x and y, we have |x + y| ≤ |x| + |y|. In this case, use a as x and -b as y.

Step 3: Substitute variables

By substituting a and -b into the theorem, we get |a + (-b)| ≤ |a| + |-b|.

Step 4: Simplify absolute values

Since the absolute value of -b is equal to the absolute value of b, this simplifies to |a| + |b|.

Step 5: Conclude the proof

Thus, we have |a - b| = |a + - b| = |a + (-b)| ≤ |a| + |b|, proving the inequality.

Key Concepts

absolute value

Understanding the concept of absolute value is essential in mathematics. The absolute value of a number is its distance from zero on the number line. This value is always non-negative, regardless of the number's sign. For instance, the absolute value of both 3 and -3 is 3. Mathematically, we denote the absolute value of a number x as \(|x|\). It helps in simplifying and working with both positive and negative numbers in equations and inequalities.
The absolute value function has the following properties that are particularly useful in proofs:

• \(|x| \geq 0\) for every real number x.
• \(|x| = 0\) if and only if x = 0.
• \(|-x| = |x|\) for every real number x.
• The Triangle Inequality: \(|a + b| \leq |a| + |b|\) for any real numbers a and b.

proofs in mathematics

Proofs are the backbone of mathematics. They provide the logical reasoning required to establish the truth of mathematical statements. In our exercise, we engage in a proof to verify the inequality \(|a - b| \leq |a| + |b|\).
Making a proof typically involves:

• Stating the theorem or problem precisely.
• Considering known results and definitions, such as those involving absolute values.
• Logically connecting steps to arrive at the conclusion.

For our problem, we use Theorem 1.2.8 as a critical step in our proof.
Keep these steps in mind when constructing any mathematical proof:

• Understand and define all terms involved.
• Break down the problem into manageable parts.
• Use previously established theorems or propositions prudently.
• Clearly show how each step leads to the next.
• Conclude cleanly, showing that you have proved the original statement.

Theorem 1.2.8

Theorem 1.2.8 is vital to solving this exercise. It states that for any real numbers x and y, \(|x + y| \leq |x| + |y|\). This theorem is crucial in the context of absolute values and inequalities, as it allows us to sum the absolute values and establish bounds.
To use Theorem 1.2.8 in our proof, we rewrite \(|a - b|\) as \(|a + (-b)|\), allowing us to apply the theorem. According to the theorem:

• Let x = a
• Let y = -b

Applying Theorem 1.2.8, we then have: \(|a + (-b)| \leq |a| + |-b|\).
We know that \(|-b|\) is the same as \(|b|\), thus simplifying our inequality to \(|a - b| \leq |a| + |b|\). This concludes our proof by demonstrating the given inequality using Theorem 1.2.8 effectively.