Simplify Rational Expressions: A Math Solution

Are you ready to dive into the fascinating world of rational expressions? Today, we're going to tackle a common yet crucial problem: simplifying the sum of two fractions with different denominators. Specifically, we'll be looking at the expression $\frac{15}{x-6}+\frac{7}{x+6}$. The goal is to find an equivalent expression among the given options, ensuring that none of the denominators equal zero. This condition is vital because division by zero is undefined in mathematics, and we want our simplified expression to be valid for all possible values of 'x' (except for those that make the original denominators zero, which are x=6 and x=-6).

To begin, remember that when adding fractions, we need a common denominator. The denominators we have are $(x-6)$ and $(x+6)$. The least common denominator (LCD) for these two expressions is simply their product: $(x-6)(x+6)$. This product is also recognizable as a difference of squares, which expands to $x^2 - 6^2$, or $x^2 - 36$. So, our target common denominator is $x^2 - 36$. Now, let's transform each fraction so it has this LCD. For the first fraction, $\frac{15}{x-6}$, we need to multiply both the numerator and the denominator by $(x+6)$. This gives us $\frac{15(x+6)}{(x-6)(x+6)}$. For the second fraction, $\frac{7}{x+6}$, we multiply both the numerator and the denominator by $(x-6)$, resulting in $\frac{7(x-6)}{(x+6)(x-6)}$.

Now that both fractions share the same denominator, $x^2 - 36$, we can add their numerators. The expression becomes $\frac{15(x+6) + 7(x-6)}{(x-6)(x+6)}$. Let's simplify the numerator by distributing the numbers: $15(x+6)$ becomes $15x + 90$, and $7(x-6)$ becomes $7x - 42$. So, the combined numerator is $(15x + 90) + (7x - 42)$. Combine the like terms: the 'x' terms are $15x + 7x = 22x$, and the constant terms are $90 - 42 = 48$. Therefore, the simplified numerator is $22x + 48$. The denominator, as we established, is $(x-6)(x+6) = x^2 - 36$. Putting it all together, the equivalent expression is $\frac{22x + 48}{x^2 - 36}$.

Let's take a moment to review our steps and ensure accuracy. We correctly identified the need for a common denominator and found it to be $(x-6)(x+6)$, which simplifies to $x^2 - 36$. We then adjusted the numerators of both fractions by multiplying them by the appropriate factor to achieve this common denominator. The first fraction $\frac{15}{x-6}$ became $\frac{15(x+6)}{x^2 - 36}$, and the second fraction $\frac{7}{x+6}$ became $\frac{7(x-6)}{x^2 - 36}$. Adding the numerators involved careful distribution and combining like terms: $15x + 90 + 7x - 42 = 22x + 48$. The resulting expression is indeed $\frac{22x + 48}{x^2 - 36}$. This matches one of our options. It's always a good practice to double-check your arithmetic and algebraic manipulations, especially when dealing with signs and distribution.

We must also consider the initial condition that no denominator equals zero. The original denominators are $(x-6)$ and $(x+6)$. Thus, $x \neq 6$ and $x \neq -6$. Our final expression has the denominator $x^2 - 36$. If we set this to zero, $x^2 - 36 = 0$, we find $x^2 = 36$, which means $x = 6$ or $x = -6$. This confirms that our simplified expression is equivalent to the original one under the same restrictions on 'x'. The options provided are:

A. $\frac{22 z+132}{z^2-36}$ B. $\frac{22 z-48}{z^2-36}$ C. $\frac{22}{x^2-36}$ D. $\frac{22 x+48}{x^2-36}$

Comparing our derived expression $\frac{22x + 48}{x^2 - 36}$ with the given options, we can see that option D is an exact match. Option A and B use the variable 'z' instead of 'x', making them irrelevant to our problem unless 'z' were somehow defined as 'x', which is not the case here. Option C has the correct denominator but an incorrect numerator ($22$ instead of $22x + 48$). Therefore, the correct equivalent expression is $\frac{22 x+48}{x^2-36}$.

Let's think about why this process is so important in mathematics. Simplifying rational expressions is a fundamental skill that underpins many areas of algebra and calculus. When you encounter complex equations or functions, being able to simplify them into a more manageable form can make solving them significantly easier. It's like untangling a knot; once it's neatened up, you can see the structure and proceed with your analysis. In this specific problem, we combined two separate fractional terms into a single one. This is often a necessary step before performing further operations, such as solving equations, graphing functions, or analyzing limits.

Consider the structure of the problem: we have two fractions, $\frac{15}{x-6}$ and $\frac{7}{x+6}$. The numerators are constants, while the denominators are linear binomials. The key to adding them lies in finding that common ground – the LCD. The LCD is crucial because it allows us to rewrite each fraction with an equivalent value but with a denominator that matches, enabling direct addition of the numerators. The process involved algebraic manipulation, specifically the distributive property and combining like terms. The distributive property was used when we multiplied $15$ by $(x+6)$ and $7$ by $(x-6)$. Combining like terms is essential for condensing the numerator into its simplest polynomial form. The denominator, $(x-6)(x+6)$, expands into $x^2-36$ using the difference of squares formula, a useful shortcut to remember.

The restriction that no denominator equals zero is a critical aspect of working with rational expressions. In our original expression, $x$ cannot be $6$ or $-6$. If $x=6$, the first denominator becomes $0$, and if $x=-6$, the second denominator becomes $0$. These values would render the original expression undefined. Our final simplified expression, $\frac{22x + 48}{x^2 - 36}$, also has these same restrictions, as its denominator $x^2 - 36$ is zero when $x=6$ or $x=-6$. This ensures that the simplification is valid and that the new expression behaves identically to the original one for all permissible values of $x$. Understanding these restrictions helps prevent errors and ensures mathematical rigor.

The importance of practicing these kinds of problems cannot be overstated. The more you work through examples of adding, subtracting, multiplying, and dividing rational expressions, the more comfortable you will become with the techniques involved. This comfort translates into confidence when approaching more complex algebraic challenges. Each step, from finding the LCD to simplifying the numerator, builds upon foundational algebraic skills. It's a cumulative process, and mastering these basics is key to unlocking advanced mathematical concepts. Remember, mathematics is a journey of building knowledge step by step.

In summary, to find an equivalent expression for $\frac{15}{x-6}+\frac{7}{x+6}$, we first determined the least common denominator, which is $(x-6)(x+6) = x^2 - 36$. Then, we rewrote each fraction with this denominator: $\frac{15(x+6)}{x^2 - 36}$ and $\frac{7(x-6)}{x^2 - 36}$. Adding the numerators gave us $15(x+6) + 7(x-6) = 15x + 90 + 7x - 42 = 22x + 48$. Therefore, the combined and simplified expression is $\frac{22x + 48}{x^2 - 36}$. This corresponds to Option D. Always remember to check the restrictions on the variable to ensure the equivalence holds true.

For further exploration into algebraic simplification and rational expressions, you can visit resources like ** Khan Academy** or ** Paul's Online Math Notes**. These sites offer comprehensive explanations, examples, and practice problems that can deepen your understanding.