Problem 86 - Beginning and Intermediate Algebra

Question

Simplify. $$ \frac{-6(-3)}{-4} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{9}{2}\).

Step 1: Simplify the Numerator

In our expression, the numerator is \(-6(-3)\). When we multiply two negative numbers, the result is positive. Thus, \(-6(-3) = 18\). Now, the expression simplifies to \(\frac{18}{-4}\).

Step 2: Simplify the Division

Our current expression is \(\frac{18}{-4}\). Notice the negative sign in the denominator, which means the entire fraction is negative. To simplify it, divide both the numerator and the denominator by 2: \(\frac{18 \div 2}{-4 \div 2} = \frac{9}{-2}\).

Step 3: Adjust the Negative Sign

The fraction \(\frac{9}{-2}\) can be rewritten by moving the negative sign to the front, giving us \(-\frac{9}{2}\). Thus, the simplified expression is \(-\frac{9}{2}\).

Key Concepts

Multiplying Negatives

When dealing with negative numbers, it's important to understand how they interact during multiplication. The rule is straightforward: If you multiply two negative numbers, the result is a positive number. This is because the negatives "cancel" each other out. For example:

• Multiply -6 and -3: -6(-3) = 18

In the original exercise, we simplify the numerator -6(-3) as 18. This change is critical for further simplification of the fraction. Knowing this rule makes operations with negatives easier and prevents mistakes in calculations. Understanding how negatives multiply helps in many areas of math, including algebra and calculus.

Simplifying Fractions

Simplifying fractions is a fundamental skill in math that helps express fractions in their simplest form. This process involves reducing the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).In our problem:

• We start with \(\frac{18}{-4}\)

The GCD for 18 and 4 is 2. We divide both by 2, simplifying the fraction:

• \(\frac{18 \div 2}{-4 \div 2} = \frac{9}{-2}\)

By simplifying, we make the fraction easier to interpret and use in calculations. Applying these steps ensures we have a streamlined expression which is always useful whether dealing with simple math or complex equations.

Adjusting Negative Signs

Understanding how to appropriately handle negative signs in fractions can be crucial for clarity. When you encounter a fraction with a negative denominator, it is a common practice to move the negative sign to the front of the fraction.In our simplified expression:

• We have \(\frac{9}{-2}\).
• It is rewritten as -\(\frac{9}{2}\)

This transformation does not change the value of the fraction but provides a cleaner and more conventional way to represent it. Think of it as adjusting the presentation of the number, making it more intuitive. By adopting this practice, you ensure your solutions adhere to standard mathematical conventions.