Problem 7 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{9}{4 x^{2}}-\frac{13}{4 x^{2}}$$

Step-by-Step Solution

Answer

The simplest form is \( \frac{-1}{x^2} \).

Step 1: Identify the Operation

The expression involves subtraction between two fractions: \( \frac{9}{4x^2} \) and \( \frac{13}{4x^2} \). Both fractions have the same denominator \(4x^2\), which simplifies the operation.

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerators directly: \( 9 - 13 \).

Step 3: Calculate the Result of the Numerator

Compute the result of the subtraction in the numerator: \( 9 - 13 = -4 \).

Step 4: Write the Resulting Fraction

Place the result from Step 3 over the common denominator: \( \frac{-4}{4x^2} \).

Step 5: Simplify the Fraction

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 4: \( \frac{-4}{4x^2} = \frac{-1}{x^2} \). The fraction is now in its simplest form.

Key Concepts

Fraction Simplification

Simplifying a fraction means reducing it to its smallest form, where the numerator and the denominator are as close to one as possible based on their greatest common divisor (GCD).
To do this, consistently follow these steps:

• First, identify the numerator and the denominator of the fraction. In our solution, it was initially \( \frac{-4}{4x^2} \).
• Next, find the GCD of the numerator and the denominator. Here, both \(-4\) and \(4x^2\) share a GCD of 4.
• Finally, divide both the numerator and the denominator by this GCD to simplify the fraction. Thus \( \frac{-4}{4x^2} \) becomes \( \frac{-1}{x^2} \).

Simplifying fractions is crucial as it makes further algebraic operations more manageable and the results more interpretable.

Subtraction of Fractions

Subtracting fractions involves a straightforward process, particularly when the denominators are the same. If two fractions have identical denominators, the subtraction process is simplified to subtracting just the numerators. Here’s how:

• The fractions \( \frac{9}{4x^2} \) and \( \frac{13}{4x^2} \) have a common denominator \(4x^2\), which simplifies our task to focus solely on subtracting the numerators.
• Subtract the second numerator from the first, leading to \( 9 - 13 = -4 \). This gives us the new numerator, \(-4\).

The key is to pay close attention to the signs, ensuring the accuracy of your subtraction. Consistently using the same denominator facilitates accurate and swift subtraction.

Common Denominators

Common denominators play a crucial role in the subtraction and addition of fractions. They ensure uniformity, allowing you to focus merely on combining or separating the numerators.

• A common denominator means you don’t need to alter the fractions before performing arithmetic operations like addition or subtraction. As in our exercise, both fractions \( \frac{9}{4x^2} \) and \( \frac{13}{4x^2} \) already have the denominator \(4x^2\), simplifying the process.
• Having a common denominator enables the subtraction or addition to occur directly in the numerators, bypassing any need for conversion or further adjustment.

When fractions do not share a common denominator, the fractions must first be converted to equivalent fractions with a common denominator. However, in our example, this step was not necessary, as they were already given in simplified form for straightforward subtraction.