Problem 27 - Calculus

Question

Show that \(|1-5|<|1|+|-5|\). Always \(\left|v_{1}+v_{2}\right|<\left|v_{1}\right|+\left|v_{2}\right|\) unless

Step-by-Step Solution

Answer

The inequality \(|1-5|<|1|+|-5|\) is always true, except when both terms are zero, as it follows the triangle inequality.

Step 1: Calculate Individual Absolute Values

First, calculate the absolute value of each term separately. For the term \(1\), we have \(|1| = 1\), and for \(-5\), we have \(|-5| = 5\).

Step 2: Calculate the Absolute Value of the Sum

Now, compute the absolute value of the sum \(1 - 5\). We get \(1 - 5 = -4\), so \(|1 - 5| = 4\).

Step 3: Compare the Absolute Values

Compare the absolute value of the sum \(|1 - 5| = 4\) with the sum of the absolute values of each term, \(|1| + | -5 | = 1 + 5 = 6\).

Step 4: Verify the Inequality

Since \(|1 - 5| = 4 \) and \(|1| + |-5| = 6\), we see that \(4 < 6\). Therefore, the inequality \(|1-5|<|1|+|-5|\) holds true.

Key Concepts

Absolute Value

The concept of absolute value is fundamental in understanding inequalities and many other mathematical areas. The absolute value of a number refers to its distance from zero on a number line, regardless of direction. It is always a non-negative number.

• If the number is positive or zero, the absolute value is itself.
• If the number is negative, the absolute value is the positive counterpart of that number.

For example, the absolute value of 1, written as \(|1|\), is 1. Similarly, the absolute value of -5, written as \(|-5|\), is 5. These calculations make it easier to compare distances and apply them in inequalities. Knowing how to compute absolute values is a vital step in proving mathematical inequalities.

Inequality

An inequality is like a handshake agreement between numbers without revealing the exact values. It tells us about the size or the range where the numbers sit relative to one another. Specifically, inequalities do not involve precise values but rather indicate whether one expression is greater than, less than, or equal to another.

• For example, the inequality \(x < y\) denotes that \(x\) is less than \(y\).
• The inequality \(a \geq b\) signifies that \(a\) is either greater than or equal to \(b\).

In the context of absolute values, inequalities become a practical tool for comparing sums of magnitudes, such as \(|1 - 5| < |1| + |-5|\). This specific inequality helps us understand how the absolute value of a sum is compared to the sum of individual absolute values.

Mathematical Proof

A mathematical proof is a logical sequence of statements that establish the truth of a proposition. It is like methodically constructing a bridge from hypothesis to conclusion with each step built solidly on the previous one. Proofs require precision and care but also creativity and insight, particularly when tackling inequalities.

• To start a proof, clearly define assumptions and known truths.
• Proceed with a series of logical reasoning steps.
• Conclude with verification that the desired outcome has been achieved.

For instance, the inequality \(|1-5|<|1|+|-5|\) is demonstrated through a proof by first evaluating each side individually and then comparing them. Such proofs are essential tools for validating mathematical statements beyond any doubt.

Vector Addition

In mathematics, vector addition refers to the process of adding two or more vectors, which has applications in physics, engineering, and more. Vectors represent quantities with both magnitude and direction.

• The outcome of vector addition is another vector.
• Visually, it corresponds to placing the tail of one vector at the tip of another.

While dealing with absolute values and inequalities, you might encounter scenarios where vector addition needs clarification, especially in geometrical interpretations. The relationship between the absolute value of a sum and the sum of absolute values often appears similar to how vectors sum in a triangle inequality context. Understanding vector addition complements interpreting inequalities involving absolute values.