Problem 38 - 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] $$\frac{3+9(-1)}{5-7}$$

Step-by-Step Solution

Answer

The expression simplifies to 3.

Step 1: Simplify the Numerator

First, focus on simplifying the expression in the numerator \(3 + 9(-1)\). According to the order of operations, perform the multiplication before addition. Thus, compute \(9 \times -1\) which equals \(-9\), so the numerator becomes \(3 + (-9)\). When adding these, you get \(3 - 9 = -6\).

Step 2: Simplify the Denominator

Next, simplify the denominator \(5 - 7\). Perform the subtraction to get \(-2\).

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now that we have a simplified numerator and denominator, perform the division. Divide \(-6\) (numerator) by \(-2\) (denominator), which yields \(3\).

Step 4: Final Simplified Expression

The simplified expression is \(3\), as obtained by dividing the results from Steps 1 and 2.

Key Concepts

Simplifying Expressions

Breaking down expressions involves managing each mathematical operation according to the correct sequence, often known as the order of operations. It all begins by focusing on smaller parts of the expression to make calculations more manageable. For example, in the expression \(3 + 9(-1)\), multiplication comes before addition.

By following the order of operations, you start by multiplying \(9\) by \(-1\) to get \(-9\). The expression in the numerator then becomes \(3 + (-9)\).

• Calculate \(3 + (-9)\).
• This results in \(-6\).

Breaking down the expression in this stepwise manner reduces errors, making complex problems simpler.

Numerators and Denominators

The numerator and denominator are the two key components of a fraction. The numerator is the top number, representing the number of parts being considered. The denominator is the bottom number, representing the total number of equal parts. Simplifying both leads to easier division.

In the expression \(\frac{3+9(-1)}{5-7}\), the calculation starts by simplifying each part separately.

• The numerator is \(3 + 9(-1)\), which simplifies to \(-6\).
• The denominator \(5 - 7\) simplifies directly to \(-2\).

By simplifying these components separately, the fraction becomes more straightforward, making the final division quicker.

Division of Integers

Dividing integers, especially with negative values, can be tricky. However, simplifying the expression previously makes it straightforward. The rule for dividing integers is simple: dividing two negative numbers or two positive numbers results in a positive number. This knowledge is vital when finding the result of a fraction.

Consider \(\frac{-6}{-2}\). Here are the steps:

• Both the numerator and the denominator are negative, so the outcome is positive.
• Divide \(6\) by \(2\) as you would with positive numbers.
• The result is \(3\).

Understanding how signed numbers interact during division is fundamental to achieving the correct results with mathematical expressions.