Problem 45 - Algebra A Combined Function

Question

Simplify. See Sections 1.5 and \(1.6 .\) \(\frac{-6-3}{2-8}\)

Step-by-Step Solution

Answer

The simplified result is \(\frac{3}{2}\).

Step 1: Simplify the Numerator

Simplify the expression in the numerator: \(-6 - 3 = -9\).Now the expression becomes: \(\frac{-9}{2 - 8}\).

Step 2: Simplify the Denominator

Simplify the expression in the denominator: \(2 - 8 = -6\).Now the expression becomes: \(\frac{-9}{-6}\).

Step 3: Simplify the Fraction

Divide both the numerator and the denominator by their greatest common divisor, which is 3.Divide -9 by 3 to get -3, and -6 by 3 to get -2. This simplifies the fraction to:\(\frac{-3}{-2}\).Since the negatives in the numerator and denominator cancel each other out, the result is:\(\frac{3}{2}\).

Key Concepts

Numerator

In a fraction, the numerator is the top number which is placed above the line. It tells you how many parts of a whole you have. In our exercise, we started by determining the simplified value of the numerator.

In the expression \(-6 - 3\), the operation is straightforward – subtracting three from negative six gives us \(-9\). This means the numerator of the fraction is \(-9\).

Remember:

• The numerator represents the count of parts considered from a total set.
• Operations such as addition and subtraction simplify this top number directly.
• Simplification might sometimes involve combining like terms or dealing with negative signs when applicable.

Having a clear numerator helps in understanding the proportion or ratio expressed by the fraction.

Denominator

The denominator is the bottom number in a fraction, which appears below the line. It indicates the total number of equal parts the whole is divided into. In the problem, you start by simplifying the denominator.

The denominator \(2 - 8\) simplifies to \(-6\). This gives us insight into how many parts we are dividing.

Key points about denominators include:

• The denominator cannot be zero, as division by zero is undefined in mathematics.
• It plays a crucial role in determining the value of the entire fraction.
• As part of simplification, ensure to maintain correct operations such as dealing with signs (positive or negative) appropriately.

Understanding the denominator allows clearer comprehension of what the fraction represents as a portion of a whole.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that divides both the numerator and the denominator without leaving a remainder. Simplifying fractions involves dividing both the numerator and the denominator by their GCD.

In our example, the GCD of \(-9\) and \(-6\) is \(3\). By dividing both by \(3\), we simplify \(\frac{-9}{-6}\) to \(\frac{-3}{-2}\). Further simplification: since both parts of the fraction are negative, they cancel out, giving us the simplest form \(\frac{3}{2}\).

Knowing about GCD helps simplify ratios effectively:

• Identifying the GCD requires finding the highest number that can evenly divide both terms.
• Simplifying using GCD results in the simplest or reduced form of a fraction.
• Understanding GCD is useful beyond fractions, in various mathematical operations and number theory.

Gaining proficiency in finding the GCD ensures comprehending and handling fractions with ease.