Problem 43 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{a+1}{2}-\frac{a-1}{2}$$

Step-by-Step Solution

Answer

The sum or difference is 1.

Step 1: Write Down the Problem

We start with the expression: \( \frac{a+1}{2} - \frac{a-1}{2} \). This is a difference of two fractions.

Step 2: Combine the Fractions

Since the fractions have the same denominator, we can combine them directly by subtracting their numerators: \( \frac{(a+1) - (a-1)}{2} \).

Step 3: Simplify the Numerator

Perform the subtraction inside the numerator: \( (a+1) - (a-1) \). Simplify this to \( a + 1 - a + 1 = 2 \).

Step 4: Write the Simplified Fraction

Substitute the simplified numerator back into the fraction: \( \frac{2}{2} \).

Step 5: Simplify the Fraction

Divide the numerator by the denominator: \( \frac{2}{2} = 1 \).

Key Concepts

Fractions

Fractions are a crucial part of algebra, and sometimes they might seem intimidating due to their abstract appearance. But in essence, fractions represent a division between two numbers: the numerator (the top part) and the denominator (the bottom part). Understanding fractions is vital because they frequently appear in algebra, whether you are combining fractions, solving equations, or working with proportions. Each fraction shows how many parts out of a whole are being considered. It's important to remember:

• A fraction can be thought of as a division operation: \(\frac{a}{b}\) means \(a\) is divided by \(b\).
• The denominator dictates into how many equal parts the whole is divided, while the numerator indicates how many of those parts are being considered.
• Fractions can be transformed into decimals through division and vice versa.

Working with fractions requires practice, especially when dealing with different operations like addition, subtraction, multiplication, and division. Luckily, the operation with fractions in this exercise was simplified by having like denominators.

Simplifying Expressions

Simplifying algebraic expressions is a fundamental concept where expressions are reduced to their simplest form. This process helps make the computations more manageable and can reveal the underlying relationships between numbers.In our exercise, simplifying the expression was key once the fractions were combined. When you simplify an expression, you focus on reducing it down without changing its value. Steps to simplify an expression generally include:

• Combining like terms: Terms that have the same variables and powers can be added or subtracted.
• Applying the distributive property: Expressions like \(a(b + c)\) are simplified to \(ab + ac\).
• Performing arithmetic carefully within parentheses.

In the solution, simplifying occurred at the point where the subtraction happened in the numerator. The practice of simplifying not only aids in calculations but also enhances your understanding of the relationships within equations.

Numerators and Denominators

Numerators and denominators are the building blocks of fractions. They define the relationship of the parts to the whole. - **Numerator**: This is the top number in a fraction. It indicates how many parts are being considered out of the whole. - **Denominator**: This is the bottom number. It tells into how many equal parts the whole is divided. In the given exercise, both fractions had the same denominator, allowing for direct subtraction of the numerators. Here, understanding the role of the numerator and denominator simplifies the process:

• When denominators are the same, arithmetic operations can be performed directly on the numerators.
• If denominators differ, you'll need to find a common denominator before performing addition or subtraction.
• Reduction of fractions occurs by simplifying the relationship between numerator and denominator through factors that they share.

Knowing these roles and operations associated with numerators and denominators allows for seamless manipulation and interpretation of fractional expressions in algebra.