Problem 31 - 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] $$\frac{2(-3)}{6-3}$$

Step-by-Step Solution

Answer

The expression simplifies to -2.

Step 1: Simplify the denominator

Look at the expression in the denominator of the fraction, which is \(6 - 3\). Simplify it by subtracting 3 from 6, resulting in 3. The expression now becomes \(\frac{2(-3)}{3}\).

Step 2: Multiply the numerator

Next, perform the multiplication in the numerator. Multiply 2 by -3, which results in -6. Now the expression is \(\frac{-6}{3}\).

Step 3: Simplify the fraction

Simplify the fraction \(\frac{-6}{3}\) by dividing -6 by 3. The result is -2.

Key Concepts

Simplifying Expressions

Simplifying expressions is a key part of algebra that helps us make complex problems easier to understand. It involves reducing expressions to their simplest form without changing their value. To simplify an expression:

• Identify and complete operations like addition, subtraction, multiplication, and division as required by the order of operations.
• Ensure all possible simplifications like factoring, canceling terms, or performing arithmetic operations are applied.

In our example, simplifying the expression \(\frac{2(-3)}{6-3}\), we identified and simplified everything both in the numerator and the denominator. By subtracting 3 from 6, we directly simplified the denominator to 3. Next, multiplying 2 with -3 simplified the numerator to -6. These steps allowed us to divide -6 by 3 easily, leading to the final result of -2. Each step ensures that the expression is as straightforward as possible before solving it completely.

Fractions

Fractions represent parts of a whole and are often involved in mathematical problems. They consist of a numerator (the top part) and a denominator (the bottom part). Understanding fractions is vital:

• The value of a fraction depends on both its numerator and denominator.
• When simplifying, it's essential to reduce fractions to their lowest terms by dividing both parts by their greatest common divisor (GCD).

In our case, \(\frac{-6}{3}\) is a fraction that can be simplified. Here, the greatest common divisor of -6 and 3 is 3. By dividing both -6 and 3 by their GCD, we simplified it to -2. Simplifying fractions helps us to handle less complicated numbers and eases further computation.

Multiplication and Division

Multiplication and division are core arithmetic operations that often appear together, especially in algebraic expressions. Knowing how and when to use these operations is crucial:

• Apply multiplication before division when working on the numerator and denominator, adhering to the order of operations.
• After multiplying, divide the resulting numbers if applicable to simplify the expression.

In the exercise \(\frac{2(-3)}{6-3}\), first we handled multiplication by calculating \(2 \times -3\), which became -6. Then, we proceeded to divide -6 by 3 to simplify the expression to -2. Balancing both operations appropriately ensures the expression is simplified accurately, which is key in algebra.