Problem 47 - Algebra A Combined Function

Question

Simplify. See Sections 1.5 and \(1.6 .\) \(\frac{-8-(-2)}{-3-(-2)}\)

Step-by-Step Solution

Answer

The simplified expression is 6.

Step 1: Distribute Minus Signs

The expression is \( \frac{-8 - (-2)}{-3 - (-2)} \). Begin by distributing the negative signs in both the numerator and the denominator: \(-8 - (-2) = -8 + 2\) and \(-3 - (-2) = -3 + 2\).

Step 2: Simplify Each Part

Simplify both the numerator and the denominator: \(-8 + 2 = -6\) and \(-3 + 2 = -1\). Your expression should now look like \( \frac{-6}{-1} \).

Step 3: Division of Negatives

When dividing two negative numbers, the result is positive. Therefore, \( \frac{-6}{-1} = 6 \).

Step 4: Final Simplification Check

Verify that the simplified expression is indeed \(6\). This is the simplest form.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero and are commonly used in mathematics to represent a deficit or opposite direction. When working with negative numbers, it is important to understand the following rules:

• Adding a negative number is equivalent to subtracting its positive counterpart, e.g., \(-8 + (-2) = -8 - 2\).
• Subtracting a negative number is akin to adding the positive version of that number, e.g., \(-8 - (-2) = -8 + 2\).

Recognizing these rules can make dealing with operations involving negative numbers much more intuitive and simpler.

Division of Negatives

Dividing negative numbers follows a straightforward rule: when two negative numbers are divided, the result is positive. This is because each pair of negative signs effectively "cancel out" each other.
For example, in the expression \(\frac{-6}{-1}\), both variables are negative. By dividing them, you end up with 6, making the result a positive number. This rule is consistent:

• \(\frac{-a}{-b} = \frac{a}{b}\)
• This holds true regardless of the values of \(-a\) and \(-b\), as long as they are the same sign.

Always ensure that when performing division with negative numbers, you check the signs to predict the outcome.

Numerator and Denominator

In a fraction, the numerator is the top part, and the denominator is the bottom part. Understanding each part is essential for performing algebraic operations efficiently.

• The numerator, in a fraction \(\frac{a}{b}\), represents how many parts we have.
• The denominator tells us how many parts make up a whole.

When simplifying fractions like \(-8 - (-2)\) over \(-3 - (-2)\), pay attention to both parts as you operate on them. Correctly addressing each term in the numerator and the denominator, such as turning \(-8 + 2 = -6\) for the numerator and \(-3 + 2 = -1\) for the denominator, ensures accuracy in simplification.

Exploring the Distributive Property

The distributive property is a useful algebraic rule that allows you to simplify expressions by distributing a multiplier over terms within parentheses. It is expressed as \(a(b + c) = ab + ac\) and it helps you to break down expressions involving negative signs.
In the context of the given exercise, consider the expression \(\frac{-8 - (-2)}{-3 - (-2)}\). The distributive property lets you change the "minus a negative" to a "plus":

• \(-8 - (-2) = -8 + 2\)
• \(-3 - (-2) = -3 + 2\)

Applying this property makes simplifying expressions less cumbersome, letting you deal with negative numbers more smoothly. It's a fundamental tool in making expressions easier to solve.