Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the indicated implication is true. $$ |x+2|<0.3 \Rightarrow|4 x+8|<1.2 $$

Step-by-Step Solution

Answer

The implication \(|x+2| < 0.3 \Rightarrow |4x+8| < 1.2\) is true, as \(4|x+2| = |4x+8| < 1.2\).

Step 1: Understand the Given Condition

The problem asks us to show that if \(|x+2| < 0.3\), then \(|4x+8| < 1.2\). We first need to understand the implication by examining the condition \(|x+2| < 0.3\). This condition tells us that the distance between \(x\) and \(-2\) is less than 0.3.

Step 2: Simplify the Implication

To prove the implication, we need to make \(|4x+8|\) in terms of \(x+2\). Notice that \(4x + 8\) can be rewritten as \(4(x+2)\). Using this, we have \(|4x+8| = |4(x+2)| = 4|x+2|\).

Step 3: Apply the Given Condition

Substitute the condition \(|x+2| < 0.3\) into the expression for \(|4x+8|\). This gives us \(4|x+2| < 4 imes 0.3 = 1.2\). Now \(|4x+8| < 1.2\).

Step 4: Conclusion

Since we have shown that the derived condition \(|4x+8| < 1.2\) directly follows from the given inequality \(|x+2| < 0.3\), the implication is true.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities are a way to express the distance of a number from zero on the real number line. These inequalities help determine a range within which a number lies. For example, \(|x+2| < 0.3\) suggests that the distance between \(x\) and \(-2\) is less than 0.3.

When working with absolute value inequalities, it's helpful to think in terms of distance:

• \(|x| < a\) translates to \(-a < x < a\).
• The inequality \(|x-a| < b\) can be expressed as \(a-b < x < a+b\), highlighting the range of \(x\) values within the bounds of \(a\) and \(b\).

To solve these inequalities, consider both the positive and negative scenarios that satisfy \(|x|\). Whether you're translating the inequality or using it to find a specific range for \(x\), understanding its geometric meaning as distance will assist in solving the problem efficiently. In the context of this exercise, substituting values helps verify that the transformed terms follow the original inequality's rules.

Logical Implications

Logical implications are statements that express a dependability from one statement (the antecedent) to another (the consequent). In simpler terms, if the antecedent is true, then logically, the consequent will also be true.

In the exercise given, the logical implication is \(|x+2|<0.3 \Rightarrow |4x+8|<1.2\). Here, our job is to prove that whenever the condition \(|x+2| < 0.3\) is satisfied, it necessarily results in the condition \(|4x+8| < 1.2\).

• The initial statement, or antecedent, is \(|x+2|<0.3\).
• The concluding statement, or consequent, is \(|4x+8|<1.2\).

Understanding logical implications requires clear identification of both components. Demonstrating an implication involves showing that the transformation or manipulation of the antecedent consistently results in the truth of the consequent. Thus, a correct transformation like replacing \(4(x+2)\) shows that implications are not just about arithmetic precision, but logical clarity.

Mathematical Proofs

Mathematical proofs are structured arguments that ensure a statement or theorem to be true. In proving implications, a common method is showing how one statement logically follows another through a series of justified steps. Each step is based on fundamental principles or previously established results.

For this problem, by transforming the expression \(|4x+8|\) in terms of \(|x+2|\), we can use a simple arithmetic method for our proof:

• Transform \(4x+8\) into \(4(x+2)\).
• Apply the condition that was initially given \(|x+2| < 0.3\) to find \(4|x+2| < 1.2\), demonstrating the required result.

The key to mathematical proofs is maintaining a clear and logical sequence. Each transition from one step to the next must be clear and justified by a rule or previously proven concept. Mathematical proofs are not about simplifying from one side to the other but about establishing a comprehensive and rigorous understanding of the relationships between elements. This methodical approach ensures reliability and accuracy in validating mathematical statements.