Problem 75 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. \(\frac{-5(-6)}{9-(-1)}\)

Step-by-Step Solution

Answer

The result is 3.

Step 1: Simplify the numerator

First, simplify the expression in the numerator. Here, you have \(-5(-6)\). Calculate the product of -5 and -6. \(-5(-6) = 30\)

Step 2: Simplify the denominator

Now, simplify the expression in the denominator. Here, you have \(9 - (-1)\). Subtracting a negative number is the same as adding the positive counterpart. \(9 - (-1) = 9 + 1 = 10\)

Step 3: Divide the results

Now, take the simplified numerator and divide it by the simplified denominator. \(\frac{30}{10} = 3\)

Key Concepts

Multiplication of Integers

Multiplication of integers involves combining two numbers to get a product. If you multiply two positive numbers or two negative numbers, the result is positive. When you multiply a positive number and a negative number, the result is negative. For example, in the problem \(-5(-6)\), you are multiplying two negative integers. According to the rule, the product is positive. Therefore, \(-5 \times -6 = 30\).
A quick check to remember:

• Positive \(\times\) Positive = Positive
• Negative \(\times\) Negative = Positive
• Positive \(\times\) Negative = Negative
• Negative \(\times\) Positive = Negative

Simplifying Expressions

Simplifying expressions involves making them easier to work with. Let's consider the denominator, where we had \(9 - (-1)\).
When you subtract a negative number, it is the same as adding the positive equivalent. So, \(9 - (-1)\) becomes \(9 + 1\).
This makes the denominator \(10\).
Simplifying expressions like this helps reduce complex operations into simpler ones. Always look for opportunities to turn subtractions involving negatives into additions.

Division of Integers

For the division of integers, you'll need to remember a few key rules:

• Positive \(\div\) Positive = Positive
• Negative \(\div\) Positive = Negative
• Positive \(\div\) Negative = Negative
• Negative \(\div\) Negative = Positive

In the problem, we ended with \(\frac{30}{10}\). Both integers are positive, so the result is also positive:
Therefore, \(\frac{30}{10} = 3\)
This basic understanding allows you to confidently handle integer division in different scenarios.