Problem 130 - Elementary and Intermediate Algebra

Question

Suppose \(a\) is a positive number and \(b\) is a negative number. Determine whether the given expression is positive or negative. a. \(-a(-b)\) b. \(\frac{-a}{b}\) c. \(\frac{-a}{a}\) d. \(\frac{1}{b}\)

Step-by-Step Solution

Answer

a. Positive; b. Positive; c. Negative; d. Negative.

Step 1: Evaluate Expression a

To determine the sign of the expression \(-a(-b)\), first calculate \(-b\), which becomes positive since \(b\) is initially negative. Therefore, \(-a(-b) = ab\). Since \(a\) is positive and \(-b\) becomes positive, the product \(ab\) is positive.

Step 2: Evaluate Expression b

To evaluate the sign of \(\frac{-a}{b}\), notice that \(-a\) is negative (since \(a\) is positive), and \(b\) is negative. Dividing a negative number by another negative number results in a positive number. Hence, \(\frac{-a}{b}\) is positive.

Step 3: Evaluate Expression c

For the expression \(\frac{-a}{a}\), \(-a\) is negative while \(a\) is positive. Dividing \(-a\) by \(a\) results in \(-1\), which is negative.

Step 4: Evaluate Expression d

The expression \(\frac{1}{b}\) involves dividing 1, a positive number, by \(b\), a negative number. This results in a negative number because the division of a positive number by a negative number is negative.

Key Concepts

Understanding Positive and Negative Numbers

When dealing with algebraic expressions, it's crucial to grasp the behavior of positive and negative numbers. Positive numbers are greater than zero, while negative numbers are less than zero.

• Positive numbers: These numbers are like +3, +5, or +10, and they move to the right of zero on a number line.
• Negative numbers: These numbers might take the form -2, -4, or -7, situated to the left of zero on a number line.

When evaluating expressions involving both, remember: - The product of two negative numbers is positive. This concept comes in handy when multiplying variables in expressions, like \(-a(-b)\). Here, \(-b\) becomes positive, resulting in a positive product. - When multiplying or dividing a positive number and a negative number together, the result is always negative.

Grasping Integer Division

Integer division involves dividing one integer by another, often producing an integer result. However, understanding how the sign of each number affects the outcome is crucial.

• Dividing a positive integer by a positive integer (e.g., \(\frac{6}{2}\)) results in a positive integer.
• Dividing a negative integer by a negative integer also results in a positive integer (e.g., \(\frac{-6}{-2} = 3\)).
• When dividing a negative integer by a positive integer, the result is a negative integer (e.g., \(\frac{-6}{2} = -3\)).
• Conversely, dividing a positive integer by a negative integer results in a negative integer (e.g., \(\frac{6}{-2} = -3\)).

In expressions such as \(\frac{-a}{b}\), where \(-a\) is negative and \(b\) is negative, the result is a positive integer because dividing two negative numbers yields a positive value.

Multiplying Integers

The process of multiplying integers revolves around understanding how the signs of the numbers involved affect the result.

• The product of a positive integer and a positive integer is always positive.
• The product of two negative integers is also positive, as they cancel out the negative signs (e.g., \(-3 \times -2 = 6\)).
• The multiplication of a positive and a negative integer, however, is negative (e.g., \(3 \times -2 = -6\)).

This understanding is key to solving expressions like \(-a(-b)\), where the double negative results in a positive product. Recognizing the patterns in these simple rules helps in quickly evaluating algebraic expressions.