Problem 63 - Precalculus

Question

Show that $$ |a+b+c| \leq|a|+|b|+|c| $$ for all real numbers \(a, b,\) and \(c .\) Hint: The left-hand side can be written \(|a+(b+c)| .\) Now use the triangle inequality.

Step-by-Step Solution

Answer

By applying the triangle inequality, we verify the inequality.

Step 1: Understand the Triangle Inequality

The triangle inequality states that for any real numbers, \(x\) and \(y\), the absolute value of their sum is less than or equal to the sum of their absolute values: \(|x+y| \leq |x| + |y|\). This property will be used to prove the given inequality.

Step 2: Apply the Triangle Inequality to Sum of b and c

Consider the expression \(|b+c|\). According to the triangle inequality, we have:\[|b+c| \leq |b| + |c|\]

Key Concepts

Absolute Value

The absolute value of a number is essentially a measure of how far that number is from zero on the number line, regardless of its direction. It's denoted as \(|x|\), where \(x\) can be any real number. The absolute value of a number is always non-negative. This is because it either retains the number itself if it's already positive or converts it to positive if it was initially negative.

• If \(x \, \geq \, 0\), then \(|x| = x\).
• If \(x \, < \, 0\), then \(|x| = -x\).

Understanding absolute value is crucial when discussing inequalities because it provides a way to express distances and sizes uniformly. When working with inequalities involving absolute values, like the triangle inequality, you're focused on comparing these sizes rather than dealing with the direction of the numbers.

Inequality Proof

Inequality proofs are important in mathematics as they establish the relative sizes of expressions without needing to know their exact values. In this exercise, you are asked to prove an inequality involving the sum of three real numbers and their absolute values. The hint suggests rewriting the expression in a specific way to aid the proof.
An inequality proof often involves several primary strategies:

• Direct application of known inequalities, such as the triangle inequality.
• Breaking down complex expressions into simpler parts.
• Using algebraic manipulations to relate the given inequality to known results.

For this specific problem, you start with the triangle inequality, which states \(|a + b| \leq |a| + |b|\). This idea is extended to the sum \(a + (b+c)\), allowing you to first apply the inequality to \(|a + (b+c)|\). Once this foundational step is broken down, it's easier to prove the larger statement \(|a+b+c| \leq |a|+|b|+|c|\)."This process of logically connecting smaller inequalities builds up to prove the original inequality concretely.

Real Numbers

Real numbers are an encompassing set that includes numbers from the domains of both rational and irrational numbers. Essentially, they cover every point on the number line.
Real numbers can be:- Positive numbers like 1, 2, and 3.- Negative numbers like -1, -2, and -3.- Zero.- Fractions and decimals like 0.5, or \(rac{3}{4}\).- Irrational numbers, which cannot be expressed as a simple fraction, like \(\pi\) or \(\sqrt{2}\).The concept of real numbers is fundamental because they provide the complete system of numbers that are used in everyday mathematics and scientific calculations. In this exercise, we're specifically working with real numbers \(a\), \(b\), and \(c\) to explore their properties through inequalities. This ensures that the statements and proofs we deal with apply broadly and generally, beyond just whole numbers. Understanding the behaviors and properties of real numbers allows for thoroughly addressing more complex mathematical topics.