Problem 82 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{-6(-3)}{-4}\)

Step-by-Step Solution

Answer

\(-\frac{9}{2}\) or \(-4.5\).

Step 1: Simplify Inside the Numerator

First, simplify the multiplication inside the numerator \(-6(-3)\). Recall that the product of two negative numbers is positive. Thus, \(-6 \times -3 = 18\). Now the expression becomes \(\frac{18}{-4}\).

Step 2: Simplify the Fraction

Simplify the fraction \(\frac{18}{-4}\). Since both 18 and 4 can be divided by their greatest common divisor, which is 2, we simplify to get \(\frac{9}{-2}\).

Step 3: Write the Fraction Properly

Since the fraction \(\frac{9}{-2}\) has a negative denominator, it can be rewritten as \(-\frac{9}{2}\) or an equivalent form with the negative sign in front: \(-4.5\).

Key Concepts

Multiplication of Integers

Multiplying integers can sometimes be confusing, but once you understand the rules, it becomes much easier. When you multiply two numbers, there are rules regarding their signs that you need to remember:

• If you multiply two positive numbers, the result is positive.
• If you multiply two negative numbers, the result is also positive. This is because the two negatives "cancel out" to make a positive.
• When you multiply a positive number with a negative number, the result is negative.

In the problem we are discussing, we have to multiply o: -6 and -3. Both numbers are negative, and according to our rules, the product of these two negative numbers is positive, which gives us 18.
This is why understanding the sign rules is so important. It helps avoid mistakes when dealing with negative numbers.

Simplifying Fractions

Simplifying fractions might sound like a daunting task, but it's really about making a fraction as simple as it can be. Here's how you can do this effectively:

• The goal is to reduce the fraction to its lowest terms. You do this by finding the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and denominator by the GCD to simplify.

For example, the fraction \(\frac{18}{-4}\) has 18 in the numerator and -4 in the denominator. The GCD of 18 and 4 is 2. Therefore, you divide 18 and -4 by 2:
\[\frac{18 \div 2}{-4 \div 2} = \frac{9}{-2}\]
By simplifying, calculations become easier, and further operations with fractions are more manageable. Understanding how to simplify fractions helps improve accuracy in math problems.

Negative Numbers in Fractions

Fractions with negative numbers can be tricky, but knowing where the negative sign should go makes a big difference. The sign can actually be placed in front of either the numerator, the denominator, or the entire fraction. Here is how it works:

• When you see a negative sign in front of the numerator, as in \(-\frac{9}{2}\), it implies the entire fraction is negative because you are essentially considering \(-9 \div 2\).
• If the negative sign is in front of the denominator, like \(\frac{9}{-2}\), you can also interpret it as \(-\frac{9}{2}\).
• Placing the negative sign in front of the whole fraction, such as \(-\frac{9}{2}\), is a common and preferred way to express a fraction with a negative sign.

Understanding that \(\frac{9}{-2}\) is equivalent to \(-\frac{9}{2}\) helps you feel more confident in rewriting and simplifying expressions. Always remember: you want the fraction to reflect the negative sign in a standard way, typically in front of the entire fraction.