Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the indicated implication is true. $$ |x+4|<\frac{\varepsilon}{2} \Rightarrow|2 x+8|<\varepsilon $$

Step-by-Step Solution

Answer

If \(|x+4|<\frac{\varepsilon}{2}\), then \(|2x+8|<\varepsilon\) is true because \(|2x+8| = 2|x+4|\).

Step 1: Understand the Implication

We need to show that when \(|x+4|<\frac{\varepsilon}{2}\) is true, it implies \(|2x+8|<\varepsilon\). This means, assuming the first inequality is true, we must logically reach the second.

Step 2: Connect the Expressions

Notice that \(2x+8 = 2(x+4)\). Therefore, \(|2x+8| = |2(x+4)| = 2|x+4|\). This crucial observation will guide our proof of the implication.

Step 3: Manipulate the First Expression

From \(|x+4|<\frac{\varepsilon}{2}\), multiply both sides by 2 to maintain the inequality, resulting in \(2|x+4| < 2 \cdot \frac{\varepsilon}{2}\). Simplifying the right-hand side gives \(2|x+4| < \varepsilon\).

Step 4: Arrive at the Desired Inequality

Since we observed that \(|2x+8| = 2|x+4|\), from Step 3 we conclude \(|2x+8| < \varepsilon\). This proves that the implication is indeed true.

Key Concepts

Absolute Value Inequalities

Understanding absolute value inequalities is essential in calculus. Absolute values determine the distance of a number from zero on the number line.
The inequality \(|x+4|<\frac{\varepsilon}{2}\) states that the distance between \(x+4\) and 0 is less than \(\frac{\varepsilon}{2}\).

• Absolute value: Represents magnitude without regard to direction.
• Inequality: Expresses the constraint or range within which a value must lie.

To manipulate absolute value inequalities, maintain balance by performing the same mathematical operations on both sides.
This is crucial to correctly solve or manipulate the inequality without altering the values or relationships.
In the given exercise, multiplying both sides of the original inequality ensures the logical transition toward the implication we wish to prove.

Logical Reasoning in Calculus

Logical reasoning in calculus involves making connections and drawing conclusions based on given premises. In this exercise, the premise is the inequality \(|x+4|<\frac{\varepsilon}{2}\).
The conclusion we want to reach is \(|2x+8|<\varepsilon\). Logical reasoning here involves understanding how multiplying an inequality affects its validity.

• A premise is a statement assumed to be true for the derivation of further conclusions.
• Logical reasoning connects premises to conclusions using valid mathematical operations.

Making a connection between \(2(x+4)\) and \(2x+8\) is a key step. It demonstrates how reasoning can transform expressions into another useful form.
This approach helps capture the transformation needed to ascertain the truth of an implication in calculus.

Inequality Manipulation

Inequality manipulation involves altering inequalities while maintaining their truths. Here, understanding equality transformation, like \(2(x+4)\), is crucial.
This approach reframes \(|x+4|<\frac{\varepsilon}{2}\) to \(|2x+8|<\varepsilon\) by simple arithmetic multiplication.

• Identifying equivalences such as \(2(x+4)\) being \(2x+8\).
• Multiplying both sides of an inequality by a positive number preserves its truth.
• Simplifying expressions on one side of the inequality can simplify or solve the equation.

In our exercise, multiplying the inequality by 2 helped transition from the initial form to the desired conclusion.
By protecting the inequality properties during manipulation, students can solve similar calculus challenges effectively.