Problem 32 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$\frac{2(-3)}{3-6}$$

Step-by-Step Solution

Answer

The expression simplifies to 2.

Step 1: Simplify the Numerator

In the numerator, we are multiplying 2 by -3: \[ 2(-3) = -6 \]So, the result of the numerator is -6.

Step 2: Simplify the Denominator

In the denominator, we perform the subtraction:\[ 3 - 6 = -3 \]Hence, the denominator simplifies to -3.

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now we divide the simplified numerator by the simplified denominator:\[ \frac{-6}{-3} = 2 \]This is because the division of two negative numbers results in a positive number.

Key Concepts

Numerator Simplification

When we talk about numerator simplification, we're focusing on simplifying the top part of a fraction. In our given expression, the numerator is \(2(-3)\). Here, multiplication takes precedence in the order of operations, so we begin by multiplying the two numbers.

In this case:

• Multiply 2 by -3.
• This results in -6 because a positive number times a negative number equals a negative number.

The calculation is straightforward if we follow the multiplication rules carefully. Simplifying the numerator makes it much easier to work on the rest of the fraction.

Denominator Simplification

The denominator simplification involves simplifying the bottom part of the fraction. In the expression \(3 - 6\), we have a simple subtraction to perform. Subtraction is the operation that we need to carry out in this context.

Here's how it is done:

• Subtract 6 from 3.
• The result is -3 because subtracting a larger number from a smaller one results in a negative.

By carefully simplifying the denominator, we resolve the fraction to a point where it is ready to be combined with the numerator. Always ensure accuracy in this step to avoid errors later in the computation.

Integer Division

Integer division comes into play when we finally divide the simplified numerator by the simplified denominator. Our expression reduces down to \(\frac{-6}{-3}\). Both the numerator and the denominator are negative in this division operation.

Here's the process:

• Divide -6 by -3.
• Since dividing two negative numbers yields a positive result, we get 2 as the answer.

Understanding integer division is crucial because it helps you predict the sign of your final answer. Logic here will tell you, two "wrongs" make a "right," meaning the negatives cancel each other out, leading to a positive quotient.