Problem 66 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Triangle Inequality and the fact that \(0<|a|<|b| \Rightarrow 1 /|b|<1 /|a|\) to establish the following chain of inequalities. $$ \left|\frac{1}{x^{2}+3}-\frac{1}{|x|+2}\right| \leq \frac{1}{x^{2}+3}+\frac{1}{|x|+2} \leq \frac{1}{3}+\frac{1}{2} $$

Step-by-Step Solution

Answer

The inequality holds: \( \left| \frac{1}{x^2 + 3} - \frac{1}{|x| + 2} \right| \leq \frac{5}{6} \).

Step 1: Applying the Triangle Inequality

We start by applying the Triangle Inequality to the expression \( \left| \frac{1}{x^2 + 3} - \frac{1}{|x| + 2} \right| \). The triangle inequality states that for any real numbers \(a\) and \(b\), \( |a - b| \leq |a| + |b| \). Hence, we can write:\[\left| \frac{1}{x^2 + 3} - \frac{1}{|x| + 2} \right| \leq \left| \frac{1}{x^2 + 3} \right| + \left| \frac{1}{|x| + 2} \right| = \frac{1}{x^2 + 3} + \frac{1}{|x| + 2}\]

Step 2: Establishing Bounds for the Expression

Next, we establish bounds for the individual fractions \( \frac{1}{x^2+3} \) and \( \frac{1}{|x|+2} \) using known inequalities. Since \(x^2 \geq 0\) and \( |x| \geq 0 \), we have:- \( x^2 + 3 \geq 3 \) implying \( \frac{1}{x^2 + 3} \leq \frac{1}{3} \)- \( |x| + 2 \geq 2 \) implying \( \frac{1}{|x| + 2} \leq \frac{1}{2} \)Thus, we can conclude:\[\frac{1}{x^2 + 3} + \frac{1}{|x| + 2} \leq \frac{1}{3} + \frac{1}{2} = \frac{5}{6}\]

Step 3: Combining the Results

Having established each part, we now can combine them into:\[\left| \frac{1}{x^2 + 3} - \frac{1}{|x| + 2} \right| \leq \frac{1}{x^2 + 3} + \frac{1}{|x| + 2} \leq \frac{5}{6}\]This shows that the original inequality holds for any \(x\).

Key Concepts

Understanding Inequality with the Triangle Inequality

Inequality rules are essential in mathematics. They tell us about the relative sizes of numbers. One powerful tool in inequalities is the Triangle Inequality. This rule states for any two real numbers \(a\) and \(b\), the absolute difference between them is less than or equal to the sum of their absolute values: \(|a - b| \leq |a| + |b|\).

This concept is like saying the direct path (a straight line) between two points is shorter or equal to any other path that you might take moving through those points, hence often named using the triangle. When applied to expressions, this becomes a handy tool to simplify and establish bounds on expressions. In our exercise, we exploited this property:

• By comparing two fractions \( \frac{1}{x^2+3} \) and \( \frac{1}{|x|+2} \).
• This allowed us to write the inequality as a sum of the fractional terms.

Hence, starting from an absolute difference inequality, we simplified the expression using the Triangle Inequality rule.

Basics of Trigonometry Principles

While the exercise doesn't explicitly delve into trigonometry, understanding this branch of mathematics adds value. Trigonometry is the study of relationships involving lengths and angles of triangles. Even though the exercise focuses on algebraic inequalities, we often need our trigonometric toolset in broader math contexts.

Trigonometry revolves around angles and sides using functions like sine, cosine, and tangent. These relate the angles of a triangle to its side lengths. In the exercise, these aren't directly involved, but consider the general relevance:

• In inequalities, understanding sine and cosine can help solve geometric angle relationships.
• In calculus, these concepts often work alongside inequalities.

Use trigonometry as a bridge to more complex concepts, enhancing your overall mathematical framework.

Establishing Fraction Bounds

Fraction bounds help us understand the limits within which a fraction lies. In the provided exercise, establishing bounds for the expression \( \frac{1}{x^2+3} \) and \( \frac{1}{|x|+2} \) was crucial.

Here's a simple breakdown:

• The term \( x^2 \geq 0 \) ensures that \( x^2 + 3 \) is always at least 3, giving \( \frac{1}{x^2 + 3} \leq \frac{1}{3} \).
• Similarly, \( |x| \geq 0 \) tells us \( |x| + 2 \) is at least 2, leading to \( \frac{1}{|x|+2} \leq \frac{1}{2} \).

The beauty of fractions is seen when setting these bounds; they give us a clear numeric limit.

To check our inequality's validity, ensuring each part lies within known fractions is key:

• Addition of the bounds \( \frac{1}{3} + \frac{1}{2} = \frac{5}{6} \) confirms the result.
• These bounds aid in painting a full picture of the mathematical rationale.

Bounds not only simplify complex expressions but confirm the order and size of terms, showing why they fit within the specified limits.