Problem 46 - Algebra A Combined Function

Question

Simplify. See Sections 1.5 and \(1.6 .\) \(\frac{4-5}{-1-0}\)

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Simplify the Numerator

The numerator of the fraction is \(4 - 5\). Since \(4 - 5 = -1\), the simplified numerator is \(-1\).

Step 2: Simplify the Denominator

The denominator of the fraction is \(-1 - 0\). Since \(-1 - 0 = -1\), the simplified denominator is \(-1\).

Step 3: Divide the Numerator by the Denominator

With the simplified expression \(\frac{-1}{-1}\), divide \(-1\) by \(-1\). The result is \(1\), since dividing a negative number by another negative number gives a positive number.

Key Concepts

Numerator and Denominator

In any fraction, we have two key parts: the numerator and the denominator, which together make up the fraction. The numerator is the number on the top of the fraction, while the denominator is the number on the bottom. Think of the numerator as how many parts we have, and the denominator as how many parts make a whole.
For the fraction \( \frac{4-5}{-1-0} \), the numerator becomes \(-1\) and the denominator also becomes \(-1\). Here are some key points to remember:

• If the numerator and denominator have the same number, like \(-1\) in this case, it simplifies to 1.
• The numerator indicates the action or the value to be divided, whereas the denominator indicates the total equal parts to divide the numerator by.
• Simplifying both helps to make calculations much simpler and more manageable.

Understanding these components are crucial first steps in working with fractions.

Negative Numbers

Negative numbers are numbers less than zero, denoted by a minus sign \((-\)). They represent values in opposite direction or deficit.
In the case of the fraction \( \frac{-1}{-1} \), both the numerator and the denominator are negative.
When dealing with calculations involving negative numbers, consider the following:

• A negative number multiplied or divided by another negative number results in a positive number.
• Adding a negative number is the same as subtraction, and subtracting a negative number is the same as addition. That's why \(4 - 5\) equals \(-1\).
• It's important to count signs. In the expression, the negative sign applies to both the \(-1\) from the numerator and \(-0\) from the denominator.

Recognizing how to handle negative numbers is vital in simplifying expressions with them.

Division of Integers

The division of integers involves determining how many times one integer can be subtracted from another. Here, the problem involved dividing the fraction \( \frac{-1}{-1} \). With fractions, dividing means determining how many groups of the denominator fit in the numerator.
Key things to consider when dividing integers:

• When dividing integers, observe the signs.
• A negative number divided by a negative number yields a positive result.
• So, for \( \frac{-1}{-1} = 1 \), since dividing the negative \(-1\) in the numerator by the negative \(-1\) in the denominator results in positive \(1\).
• This situation flips the signs, ensuring a positive outcome.

Understanding these fundamentals makes it easier to handle expressions where you're required to divide integers.