Problem 4

Question

Add. Simplify your answer. $$ \frac{x}{x^{2}-9}+\frac{3 x+1}{x^{2}-9} $$

Step-by-Step Solution

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Answer

The simplified sum of the expressions is $ \frac{4x+1}{x^{2}-9} $.

1Step 1: Identify Common Denominator

The common denominator of the two fractions is $ x^{2}-9 $

2Step 2: Combine the Numerators

Add together the two numerators: $ x $ and $ 3x+1 $. This gives: $ x+(3x+1)= 4x+1 $

3Step 3: Write Combined Fraction

Place the combined numerators over the common denominator to get the sum of the fractions: $ \frac{4x+1}{x^{2}-9} $

Key Concepts

Common DenominatorCombining NumeratorsSimplifying Fractions

Common Denominator

In the world of algebraic fractions, a common denominator is essential when working with sums or differences. Imagine two fractions that you wish to combine; they must share the same base - or denominator - before you can proceed with addition or subtraction.

A common denominator is like the canvas both fractions need to paint on. Without it, you can't blend them together smoothly.

For example, consider the fractions $ \frac{x}{x^2-9} $ and $ \frac{3x+1}{x^2-9} $.
Notice both denominators are $ x^2 - 9 $. This simplifies your work drastically since you already have a common denominator.

Identifying same denominators is the foundation of dealing with fractions. Once you've verified that fractions share this key component, you can move on to the next step.

Combining Numerators

Now that we have a common denominator, the fun part begins: combining numerators. This is exactly what it sounds like - you simply add or subtract the top parts of the fractions while keeping the denominator unchanged.

In the given exercise, look at the two numerators:

First numerator is $ x $.
Second numerator is $ 3x + 1 $.

Here's what you do:

Add the numerators: $ x + (3x + 1) = 4x + 1 $.

Remember to focus on combining like terms. If both numerators contain variables, collect them together. Do the same for constant numbers. This ensures the numerator remains neat and tidy, setting up the fraction for any further simplification.

Simplifying Fractions

You've reached the last leg of the journey - simplifying the fraction. Imagine it like cleaning up after cooking; it's important to tidy up to highlight your fine result.

Your combined fraction is $ \frac{4x+1}{x^2-9} $.

To simplify, check if there's a common factor between the numerator and the denominator.

If a common factor exists, factor it out to shrink the fraction down as much as possible.
In this example, the numerator $ 4x + 1 $ has no common factors with the denominator $ x^2-9 $, so the fraction is already in its simplest form.

By simplifying, you ensure that your answer is not just correct, but also as concise as possible. This makes interpreting and using the solution easier for everyone involved.

Problem 3

Problem 4

Other exercises in this chapter

Problem 3

Simplify the expression. $$\frac{3 x}{8 x^{2}} \cdot \frac{4 x^{3}}{3 x^{4}}$$

View solution

Problem 3

Define the simplest form of a rational expression. Give an example of a rational expression in simplest form.

View solution

Problem 4

In Exercises 3–6, simplify the expression. $$ \frac{3}{10 x}-\frac{1}{4 x^{2}} $$

View solution

Problem 4

Find the least common denominator. $\frac{3}{4 x}, \frac{1}{6 x^{2}}, \frac{1}{8 x^{2}}$

View solution