Problem 70 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show each of the following: (a) \(x1\) (b) \(x^{2}

Step-by-Step Solution

Answer

(a) True for \(x<0\) or \(x>1\). (b) True for \(0

Step 1: Analyze Inequality for x < 0

Consider the inequality \( x < x^2 \). When \( x < 0 \), the inequality must hold. Since any negative number squared results in a positive value (and is greater than the negative number itself), \( x < x^2 \) is true for \( x < 0 \).

Step 2: Analyze Inequality for x > 1

Next, consider the inequality \( x < x^2 \) for \( x > 1 \). In this range, each number is greater than one, and squaring it makes it even larger. Thus, the inequality \( x < x^2 \) holds true for every \( x > 1 \).

Step 3: Derive Condition for x < x^2

Combine the observations from Steps 1 and 2. The combined ranges where \( x < x^2 \) is valid are \( x < 0 \) or \( x > 1 \). This meets the conditions given in part (a).

Step 4: Analyze Inequality for 0 < x < 1

Consider the inequality \( x^2 < x \). When \( 0 < x < 1 \), squaring any number in this interval results in a smaller number (since \( x \times x < x \)). Hence, \( x^2 < x \) is true for \( 0 < x < 1 \).

Step 5: Confirm Conditions for Both Inequalities

For part (a), \( x < x^2 \) holds for \( x < 0 \) and \( x > 1 \). For part (b), \( x^2 < x \) holds for \( 0 < x < 1 \). All specified conditions are confirmed to be true.

Key Concepts

Squaring Negative Numbers

When working with inequalities involving squaring, it's essential to understand what happens when you square negative numbers.

• When a negative number is squared, the result is positive.
• The absolute value of the squared result is greater than the original negative number.

For example, if you have a negative number like \(-3\), squaring it gives \((-3)^2 = 9\). This property means that for any negative \(x\), it will always hold true that \(x < x^2\).This understanding is crucial when dealing with inequalities, as it gives us a clear direction on how values behave when squared, especially those less than zero.

Real Number System

The real number system consists of numbers we commonly use in everyday life including whole numbers, fractions, and decimals. It includes certain important subsets:

• Natural numbers (1, 2, 3, ...).
• Whole numbers (0, 1, 2, 3, ...).
• Integers (..., -2, -1, 0, 1, 2, ...).
• Rational numbers (fractions like 1/2, 3/4).
• Irrational numbers (like \(\pi\), which cannot be exactly expressed as a fraction).

Understanding this system is important in analyzing inequalities because it helps categorize numbers and determine properties each group holds. For example, natural numbers behave predictably as they grow, which is useful when examining how operations like squaring affect inequality relationships.

Comparing Inequalities

When comparing inequalities such as \(x < x^2\) for different ranges of \(x\), identifying the valid intervals is essential. We consider two main intervals for these comparisons:

• For \(x < x^2\), the inequality holds when \(x < 0\) and \(x > 1\).
• For \(x^2 < x\), the inequality is true when \(0 < x < 1\).

By considering the behavior of numbers in these intervals, we can understand why they satisfy certain inequalities. For instance, when \(x > 1\), each squaring operation increases the size of the number significantly, thus making \(x < x^2\) true. Similarly, squaring numbers between 0 and 1 diminishes their size, making \(x^2 < x\) valid.These considerations allow deeper comprehension of inequalities, ensuring an accurate analysis of numerical relationships.

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Key Concepts

Squaring Negative Numbers

When working with inequalities involving squaring, it's essential to understand what happens when you square negative numbers.

• When a negative number is squared, the result is positive.
• The absolute value of the squared result is greater than the original negative number.

For example, if you have a negative number like \(-3\), squaring it gives \((-3)^2 = 9\). This property means that for any negative \(x\), it will always hold true that \(x < x^2\).This understanding is crucial when dealing with inequalities, as it gives us a clear direction on how values behave when squared, especially those less than zero.

Real Number System

The real number system consists of numbers we commonly use in everyday life including whole numbers, fractions, and decimals. It includes certain important subsets:

• Natural numbers (1, 2, 3, ...).
• Whole numbers (0, 1, 2, 3, ...).
• Integers (..., -2, -1, 0, 1, 2, ...).
• Rational numbers (fractions like 1/2, 3/4).
• Irrational numbers (like \(\pi\), which cannot be exactly expressed as a fraction).

Understanding this system is important in analyzing inequalities because it helps categorize numbers and determine properties each group holds. For example, natural numbers behave predictably as they grow, which is useful when examining how operations like squaring affect inequality relationships.

Comparing Inequalities

When comparing inequalities such as \(x < x^2\) for different ranges of \(x\), identifying the valid intervals is essential. We consider two main intervals for these comparisons:

• For \(x < x^2\), the inequality holds when \(x < 0\) and \(x > 1\).
• For \(x^2 < x\), the inequality is true when \(0 < x < 1\).

By considering the behavior of numbers in these intervals, we can understand why they satisfy certain inequalities. For instance, when \(x > 1\), each squaring operation increases the size of the number significantly, thus making \(x < x^2\) true. Similarly, squaring numbers between 0 and 1 diminishes their size, making \(x^2 < x\) valid.These considerations allow deeper comprehension of inequalities, ensuring an accurate analysis of numerical relationships.