Problem 68 - Algebra A Combined Function

Question

Perform each indicated operation. See Section R .2. $$ -\frac{3}{2}+\left(\frac{1}{2}-\frac{3}{2}\right) $$

Step-by-Step Solution

Answer

The result is \(-\frac{5}{2}\).

Step 1: Simplify the Expression Inside the Parentheses

First, look inside the parentheses: \( \left( \frac{1}{2} - \frac{3}{2} \right) \). To subtract these fractions, they must have a common denominator. Both fractions already have the same denominator (2), so simply subtract the numerators: \( 1 - 3 = -2 \). Thus, \( \frac{1}{2} - \frac{3}{2} = -\frac{2}{2} = -1 \).

Step 2: Plug Back the Result Into the Expression

Replace the expression inside the parentheses with the result from Step 1. The expression now becomes \( -\frac{3}{2} + (-1) \).

Step 3: Simplify the Expression Without Parentheses

Rewrite \( -\frac{3}{2} + (-1) \) as \( -\frac{3}{2} - 1 \). Now, convert \(-1\) to a fraction with the same denominator as \(-\frac{3}{2}\). This converts \(-1\) to \(-\frac{2}{2}\).

Step 4: Add the Fractions

Combine the fractions: \( -\frac{3}{2} - \frac{2}{2} \). Since the denominators are the same, add the numerators: \(-3 - 2 = -5 \). Therefore, \(-\frac{3}{2} - \frac{2}{2} = -\frac{5}{2} \).

Key Concepts

Common Denominator

When working with fractions, having a common denominator is crucial before performing any arithmetic operations like addition or subtraction. It ensures that both fractions have the same ‘size’ of pieces and allows us to directly compare or combine them.

• The denominator is the bottom number of a fraction, representing how many equal parts the whole is divided into.
• A common denominator means both fractions are broken down into the same number of parts, making it straightforward to manage their numerators.

In the given problem, \[\left( \frac{1}{2} - \frac{3}{2} \right)\]both fractions already share a common denominator of 2, simplifying the process since you can immediately proceed to subtract the numerators.

Numerator Subtraction

After confirming a common denominator, the next step is to subtract the numerators. The numerator is the top part of the fraction, indicating how many portions you have.

• It's like subtracting two numbers while keeping the bottom part (denominator) constant.
• This requires only basic subtraction, so it's an uncomplicated arithmetic step once you have a common denominator.

In our exercise, we need to subtract the numerators within the fraction in the parentheses:\[1 - 3 = -2\]Thus, \[\frac{1}{2} - \frac{3}{2} = -\frac{2}{2} = -1\]The result is a simplified whole number after dividing the numerator by the denominator.

Parentheses Simplification

Parentheses in mathematical expressions indicate parts of the equation that should be solved first. This is an essential rule of arithmetic, often involving simplifying complex expressions to make them more manageable.

• Begin by focusing on what’s inside the parentheses.
• This often involves performing basic operations like addition or subtraction.

For the task at hand, the expression \[\left( \frac{1}{2} - \frac{3}{2} \right)\]is simplified to -1. Once simplified, you can rewrite the entire expression without parentheses by substituting this value back, which results in a much clearer and easier-to-solve equation: \[-\frac{3}{2} + (-1)\]

Negative Numbers

Negative numbers can alter the direction or result of an expression significantly. They denote values less than zero and follow specific rules in arithmetic. Understanding how to work with them is crucial:

• Negatives switch signs when subtracted, for instance, changing + to -.
• Combining negatives requires careful attention to ensure correct operations.

In this particular problem, negative numbers play an apparent role:\[-\frac{3}{2} + (-1)\]This expression is redrawn as \[-\frac{3}{2} - 1\]Here, you combine the values by turning negative addition into straightforward subtraction. Subsequently, resolving these gives \[\frac{-3}{2} - \frac{2}{2} = \frac{-5}{2}\]It's crucial to note how negative signs impact the operation and the final result, reinforcing the significance of mastering negative number arithmetic.