Problem 35 - Prealgebra

Question

Add or subtract as indicated. $$3-\frac{2}{3 x}$$

Step-by-Step Solution

Answer

The result is \(\frac{9x - 2}{3x}\).

Step 1: Identify the Operation

The expression you are given is \( 3 - \frac{2}{3x} \). Here, you need to perform subtraction.

Step 2: Find Common Denominator

In order to subtract \(3\) and \(\frac{2}{3x}\), you need a common denominator. The first term \(3\) can be written as \(\frac{3}{1}\). The denominators are \(1\) and \(3x\). The least common denominator (LCD) is \(3x\).

Step 3: Rewrite Using Common Denominator

Convert \(3\) into a fraction with denominator \(3x\) by multiplying both the numerator and the denominator by \(3x\): \[3 = \frac{3 \times 3x}{1 \times 3x} = \frac{9x}{3x} \].Now you have \(\frac{9x}{3x} - \frac{2}{3x}\).

Step 4: Subtract the Fractions

With a common denominator, subtract the fractions:\[\frac{9x}{3x} - \frac{2}{3x} = \frac{9x - 2}{3x}\].

Step 5: Simplify the Expression if Possible

The fraction \(\frac{9x - 2}{3x}\) cannot be simplified further because \(9x - 2\) and \(3x\) have no common factors other than 1.

Key Concepts

Least Common Denominator

When working with fraction subtraction, it's crucial to identify a common denominator, known as the Least Common Denominator (LCD). In this exercise, when we look to subtract the whole number 3 from the fraction \( \frac{2}{3x} \), we notice that 3 can be expressed as a fraction \( \frac{3}{1} \).
The denominators here are 1 and \( 3x \). To find the LCD, determine the smallest multiple that these denominators share. Since the denominator \( 3x \) is already larger than 1 and accommodates both constants and variables, it naturally becomes the LCD for both fractions.
By converting both fractions to have this common denominator, you ensure that their subtraction or addition is legitimate, maintaining proportionality between them.
Once you establish \( 3x \) as the LCD, you can effectively manipulate the fractions to prepare them for subtraction.

Fraction Subtraction

Subtracting fractions involves more than just working with numbers alone. First, you must ensure both expressions have a common denominator, allowing you to directly subtract the numerators. Let's break it down further:
1. **Prepare the fractions**: Convert each term to have the same denominator. In our exercise, we convert the whole number 3 to the fraction \( \frac{9x}{3x} \), using the LCD \( 3x \).
2. **Direct subtraction**: Now, with \( \frac{9x}{3x} - \frac{2}{3x} \), you can subtract the numerators directly because they share a common denominator. The subtraction is simple: \( 9x - 2 \).
3. **Maintain the denominator**: The denominator \( 3x \) remains unchanged after subtraction, resulting in \( \frac{9x - 2}{3x} \).
By understanding these steps, fraction subtraction becomes more manageable, making complex expressions less intimidating.

Algebraic Expressions

Algebraic expressions involve variables and constants, and operations such as addition and subtraction. In this exercise, you manage both a numerical constant and an algebraic fraction.
When dealing with the expression \( 3 - \frac{2}{3x} \), recognize that it combines a whole number and an algebraic fraction. Hence, understanding how to manipulate algebraic expressions is key:
- **Convert constants to fractions**: Express the constant term (3 in this case) in terms of a fraction that shares the algebraic denominator. This creates compatible expressions for operations.
- **Balance the equation**: Use algebraic principles, like keeping expressions balanced and combining like terms (in this case, dealing with the numerators).
- **Simplify**: While the algebraic expression \( \frac{9x - 2}{3x} \) looks complex, it's about combining elements logically, ensuring constants and variables are appropriately managed.
Through such manipulation, algebraic expressions simplify to more understandable forms, aiding in solving equations involving both numerical and variable components.