Problem 25 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{3}{4 x}+\frac{5}{4 x} $$

Step-by-Step Solution

Answer

Question: Add the rational expressions $$\frac{3}{4x}$$ and $$\frac{5}{4x}$$, then simplify the result. Answer: $$\frac{8}{4x}$$

Step 1: Identify the Common Denominator

Since the denominators of both fractions are already the same (4x), there is no need to find the least common denominator (LCD). We can proceed with the addition.

Step 2: Add the Numerators

To add the expressions, we keep the common denominator and add the numerators: $$ \frac{3}{4x} + \frac{5}{4x} = \frac{3 + 5}{4x} $$

Step 3: Simplify the Result

Now, we just need to simplify the numerator: $$ \frac{3+5}{4x} = \frac{8}{4x} $$ The fraction is now completely simplified, and we have our final answer: $$ \frac{8}{4x} $$

Key Concepts

Adding Rational Expressions

Adding rational expressions follows a process similar to adding regular fractions. When adding fractions, the denominators must match. Rational expressions are just fractions where the numerator and denominator are polynomials. Here's how you can add them efficiently:- Check the denominators of both rational expressions. If they are the same, great! You can proceed by adding the numerators directly.- If they are different, you'll need to find a common denominator. However, in exercises like the one stated, where the denominators are already the same, it's a simple matter of directly summing the numerators over the common denominators.In our example, adding \[\frac{3}{4x} + \frac{5}{4x}\]we notice that the denominators are identical, which makes the process straightforward. Simply add the numerators 3 and 5 to get the resulting rational expression. Understanding this basic principle can save you a lot of time and help avoid unnecessary mistakes.

Common Denominator

The common denominator is a key concept when dealing with fractions, including rational expressions. It refers to a shared multiple of the denominators of two or more fractions, allowing them to be added or subtracted easily. The steps to find it are as follows:- Identify the denominators of your rational expressions.- If the denominators are the same, you are in luck—like in our problem, where both denominators are \(4x\).- When they are not the same, you need to find a common denominator, often the least common denominator (LCD), which is the smallest possible shared multiple.A common denominator allows you to combine numerators while preserving the harmony of the fraction structure. It plays a crucial role in simplifying the calculations and ensuring you are working within the framework of valid mathematical operations. By mastering finding common denominators, you'll build a solid foundation for more complex algebraic operations.

Simplifying Fractions

Simplification of fractions is the process of reducing them to their simplest form while maintaining the same value. For rational expressions, this often involves both the numerator and the denominator.To simplify a fraction:- Simplify the numerator if possible by combining like terms or performing arithmetic operations.- Check if the resulting expression can be reduced further by canceling common factors in the numerator and denominator.In the context of our example:\[\frac{8}{4x}\]we see that both 8 and 4 can be divided by 4. This allows us to reduce the fraction:\[\frac{8}{4x} = \frac{2}{x}\]Simplifying fractions is an essential skill that not only makes your solutions clearer but also more elegant and manageable. Always look for opportunities to simplify wherever possible to make your calculations as efficient as possible.