Problem 40 - Intermediate Algebra

Question

Perform the operations. See Examples 4 and 5 . $$ \frac{16}{-4} $$

Step-by-Step Solution

Answer

\( \frac{16}{-4} = -4 \).

Step 1: Identify the Numbers and Sign

In the expression \( \frac{16}{-4} \), we have two numbers: 16 in the numerator (the top of the fraction) and -4 in the denominator (the bottom of the fraction). The operation involved is division, and notice that the denominator is negative.

Step 2: Simplify the Absolute Values

First, simplify the absolute values of the numbers. The absolute value of 16 is 16, and the absolute value of -4 is 4. Hence, focusing on the numbers themselves, we compute \( 16 \div 4 = 4 \).

Step 3: Determine the Sign of the Result

Since 16 is positive and -4 is negative, the division of these two numbers will result in a negative number. This is a rule of division: dividing numbers with different signs results in a negative number. Therefore, the result is -4.

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering the direction (negative or positive). Simply put, the absolute value measures how far a number is from zero, always yielding a non-negative result. For example, both +3 and -3 have an absolute value of 3.

When dealing with division or multiplication, it's essential to first consider the absolute values of the numbers involved. This allows us to focus on their size before addressing the signs. For instance, in the division \( \frac{16}{-4} \), we first simplify using absolute values: \( \frac{|16|}{|4|} = \frac{16}{4} = 4 \).

• Absolute value of a positive number is the number itself.
• Absolute value of a negative number is the positive counterpart of that number.

Recognizing absolute values simplifies calculations and helps ensure accuracy before considering signs.

Working with Negative Numbers

Negative numbers are less than zero and are often represented with a minus sign (-). They are a fundamental part of mathematics, appearing in everyday life while dealing with temperatures, debts, and elevations below sea level. In operations such as division, understanding how negative numbers interact is crucial.

When dividing negative numbers by positive numbers, as in \( \frac{16}{-4} \), it's important to remember these rules:

• When a positive number is divided by a negative number, the result is negative. This is because they have different signs.
• Inversely, when a negative number is divided by a positive one, or vice versa, the result remains negative.
• If both numbers in the division are negative, the result becomes positive, e.g., \( \frac{-16}{-4} = 4 \).

By applying these principles, you can accurately determine the signs of your results in operations involving negative numbers.

Operations with Fractions

Fractions represent division between two numbers, where the numerator (top number) is divided by the denominator (bottom number). When simplifying fractions, particularly those with negative values, there's a straightforward approach to ensure a correct outcome.

To simplify fractions like \( \frac{16}{-4} \), follow these steps:

• First, consider the absolute values of both parts and perform the division: \( 16 \div 4 = 4 \).
• Next, determine the sign of the result: because one number (the denominator in this case) is negative, the outcome of the division is negative, resulting in \(-4\).

Understanding these operations simplifies dealing with fractions significantly. Whenever you encounter fraction operations involving negatives, remember to assess and simplify the absolute values before applying sign rules.