Problem 81 - Calculus with Concepts in Calculus

Question

Use the definition of absolute value to prove the following. a. \(|a b|=|a||b|\) b. \(-|b| \leq b \leq|b|\) c. \(|a-b|=|b-a|\)

Step-by-Step Solution

Answer

The statements hold true based on absolute value properties.

Step 1: Recall the Definition of Absolute Value

The absolute value of a number \( x \), denoted as \( |x| \), is defined as the non-negative value of \( x \) without regard to its sign. It is expressed as: \( |x| = x \) if \( x \geq 0 \) and \( |x| = -x \) if \( x < 0 \).

Step 2: Prove Part a) |ab| = |a||b|

For any real numbers \( a \) and \( b \), consider the different possible sign combinations for \( a \) and \( b \).- If both \( a \) and \( b \) are non-negative or both are negative, \( |a| = a \) and \( |b| = b \). Thus, \( |ab| = ab = |a||b| \).- If one of \( a \) or \( b \) is positive and the other is negative, \( |ab| = |a||b| \) because \(-ab\),\(-a|b| = |a||b|, -b|a|=|a||b|\).In each case, \( |ab| = |a||b| \) holds true by definition.

Step 3: Prove Part b) -|b| ≤ b ≤ |b|

By definition, \(|b| = b\) if \(b \geq 0\) and \(|b| = -b\) if \(b < 0\), which implies that \(|b| \geq b\) in both cases.- If \(b \geq 0\), \(|b| = b\) and \(-|b| = -b\), so \(-b \leq b\), thus fulfilling \(-|b| \leq b \leq |b|\).- If \(b < 0\), \(-b \leq 0\) so \(-|b| = b \geq b\), still satisfying the inequality.

Step 4: Prove Part c) |a-b| = |b-a|

Observe that \(|a-b| = |-(b-a)|\). Using the property of absolute values \(|-x| = |x|\), it follows that \(|-(b-a)| = |b-a|\). Therefore, \(|a-b| = |b-a|\) by the definition of absolute value.

Key Concepts

Definition of Absolute Value

The concept of absolute value is foundational in mathematics. Understanding it helps to tackle a variety of problems, especially those involving distance and inequality.

Absolute value is denoted by vertical bars around a number, such as \(|x|\). It refers to the number's distance from zero on the number line, ignoring any negative sign. This means:

• \(|x| = x\) if \(x \geq 0\): When \(x\) is zero or positive, its absolute value is simply \(x\) itself, as there's no need to adjust for sign.
• \(|x| = -x\) if \(x < 0\): For a negative \(x\), its absolute value is \(-x\) because we want the positive equivalent, the distance from zero.

Think of absolute value as a way to express 'how far' a number is from zero, irrespective of its direction on the number line. This concept is essential when considering problems dealing with distance and magnitude.

Properties of Absolute Value

Once you understand what absolute value represents, it's vital to know its properties, which help in simplifying expressions and solving equations. Let's explore some key properties:

• Multiplicative Property: \(|ab| = |a||b|\) shows that the absolute value of a product is equal to the product of the absolute values. This holds regardless of the signs of \(a\) and \(b\), as absolute value eliminates any negative aspects.
• Non-negativity: \(|x| \geq 0\) means the absolute value of any real number is non-negative. It cannot be negative since distance can't be negative.
• Identity: \(|x| = 0\) if and only if \(x = 0\). This indicates that the only number with zero absolute value is zero itself.

Understanding these properties aids in navigating problems involving absolute value, particularly when working with equations, multiplication, and simplifications. They highlight the versatile nature of absolute values in handling real numbers effectively.

Inequalities involving Absolute Value

Inequalities with absolute values can initially seem daunting, but breaking them down simplifies the process. They often relate to distance and involve a range of solutions.

Consider the inequality \(-|b| \leq b \leq |b|\):

• This expresses that any real number \(b\) is always between its negative absolute value and positive absolute value.
• The reason is, regardless of \(b\)'s sign, its maximum distance from zero, which is \(|b|\), will always envelope \(b\) itself.

To solve inequalities involving absolute values:

• Think about distance: The inequality \(|x - a| < b\) implies values of \(x\) are within \(b\) units of \(a\). It’s about maintaining a certain "closeness."
• Break into cases: Solve using the definition of absolute value, considering both the positive and negative scenarios.

These strategies simplify working with inequalities and highlight how absolute value reflects distance, which is central to grasping these expressions.