Problem 3 - Contemporary Precalculus

Question

\(b, c,\) and \(d\) are real numbers such that \(b<0\) \(c>0,\) and \(d<0 .\) Determine whether the given number is positive or negative. $$-b$$

Step-by-Step Solution

Answer

Explain how you determine this. Answer: The number \(-b\) is positive since \(b\) is a negative number and negating a negative number results in a positive number.

Step 1: Recall the condition about b

We are given that \(b\) is a negative number because \(b < 0\).

Step 2: Negate a negative number

When negating a negative number, the result becomes positive. Mathematically, this can be stated as \((-1) \times b = -b\).

Step 3: Determine the sign of -b

Since \(b\) is negative and we are negating it, \(-b\) is a positive number.

Key Concepts

Negation of Numbers

When dealing with real numbers, understanding negation is key. Negation involves changing the sign of a number. Suppose you start with a negative number, such as \(b\) where \(b < 0\). Negating \(b\) means you multiply it by \(-1\). This effectively flips its position on the number line and makes it positive.

For example, if \(b = -5\), then \(-b = -(-5) = 5\). The outcome of negating any negative number is a positive number.

• Negation flips the number's sign.
• A negative number becomes positive.
• A positive number becomes negative.

This basic operation is crucial in solving equations and manipulating expressions.

Sign Determination

Determining the sign of a number involves understanding whether it is positive or negative. Let's walk through how to determine the sign of \(-b\), knowing \(b < 0\).

• \(b < 0\) indicates that \(b\) is a negative number.
• When you take the negative of a negative number, the result is positive.
• Therefore, \(-b\) is positive.

The rule is simple:

• If you negate a negative, it becomes positive.
• If you negate a positive, it becomes negative.

Understanding these sign changes is fundamental in mathematics, especially when solving inequalities or evaluating expressions.

Number Properties

Real numbers have distinct properties that help us understand and manipulate them. Some essential properties when dealing with real numbers include:

• Closure: Real numbers are closed under addition, subtraction, and multiplication, meaning any operation with real numbers yields another real number.
• Commutative Property: Order doesn't matter for addition and multiplication (e.g., \(a + b = b + a\)).
• Associative Property: Grouping doesn't affect addition and multiplication (e.g., \((a + b) + c = a + (b + c)\)).
• Distributive Property: You can spread multiplication over addition or subtraction (e.g., \(a(b+c) = ab + ac\)).

These properties ensure that no matter how numbers are combined, the basic rules remain consistent. They also help in simplifying complex expressions, making calculations more straightforward.