Problem 6 - Algebra 1

Question

Simplify the expression. $$\frac{x}{x^{2}-9}-\frac{3 x+1}{x^{2}-9}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(-\frac{2x+1}{x^{2}-9}\).

Step 1: Recognize common denominators

Two fractions can be combined or subtracted if they share the same denominator. Here, both fractions have \(x^{2}-9\) as the denominator.

Step 2: Combine the fractions

Subtract the second fraction from the first to combine them into a single fraction. This gives us \((x - (3x+1)) / (x^{2}-9)\) which simplifies to \((-2x -1) / (x^{2}-9)\).

Step 3: Simplify the expression

Ensure that the expression is in its simplest form. As there are no common factors between the numerator and the denominator, the expression \(-2x -1 / x^{2}-9\) is the final simplified form.

Key Concepts

Understanding a Common Denominator

When dealing with fractions, a common denominator is crucial for performing operations like addition and subtraction. It represents a shared base for the fractions, allowing them to be compared and manipulated easily. If our fractions have the same denominator, we can seamlessly combine or subtract them without any extra steps. In the given exercise, both fractions have a denominator of \(x^2 - 9\), making it straightforward to subtract them. By recognizing the shared \(x^2 - 9\), we bypass any necessity for adjustment and can proceed directly to subtraction. It’s important to remember that checking for a common denominator should be your initial step when performing operations with fractions. This forms the foundation for any further simplification. Without aligning the denominators, combining or reducing fractions isn't possible.

Subtracting Fractions

Subtracting fractions can seem tricky at first, but with a shared denominator, it’s simply a matter of subtracting the numerators. Think of it like subtracting two apples from three apples when the apples (or, in our case, denominators) are of the same type. To subtract fractions:

• Ensure the denominators are identical.
• Subtract the numerator of the second fraction from the numerator of the first.
• Keep the common denominator the same.

In our example, the subtraction takes place in the numerators alone: \[(x) - (3x + 1)\]which combines to form the numerator:\[-2x - 1.\]This subtraction has no effect on the denominator \(x^2 - 9\), so it remains unchanged and takes us closer to the simplifying stage.

Simplifying Rational Expressions

Once you have subtracted the fractions, you'll want to simplify the resulting fraction to its simplest form, where possible. Simplification involves reducing both the numerator and the denominator by any common factors, akin to simplifying numerical fractions. In our case, the expression \[\frac{-2x - 1}{x^2-9}\]is already in its simplest form. This is because the numerator \(-2x - 1\) and the denominator \(x^2 - 9\) do not share any common factors apart from ±1, and further reduction isn't possible."Simplifying" can involve factoring, canceling terms, or recognizing special algebraic identities such as difference of squares. Even when simplification can't reduce further, confirming that no factors remain ensures the integrity of your answer. It also helps you confirm that you have reached the simplest and most elegant form of the expression.