Simplifying $\frac{5x}{x-4}-\frac{-2x+3}{2x^2-11x+12}$ A Step-by-Step Guide

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Hey guys! Today, we're diving deep into a fascinating algebraic expression: $\frac{5x}{x-4} - \frac{-2x+3}{2x^2 - 11x + 12}$. This might look intimidating at first glance, but don't worry, we'll break it down step by step, making it super easy to understand. Think of it as solving a puzzle – each step is a piece that fits perfectly to reveal the final solution. We will explore how to simplify this expression, touching on key concepts like factoring, finding common denominators, and combining like terms. So, grab your metaphorical math hats, and let's get started on this exciting journey!

Understanding the Expression

Before we jump into the nitty-gritty, let's take a moment to understand what we're working with. The expression $\frac{5x}{x-4} - \frac{-2x+3}{2x^2 - 11x + 12}$ involves two fractions. The first fraction is $\frac{5x}{x-4}$, which is a relatively simple rational expression. The second fraction, $\frac{-2x+3}{2x^2 - 11x + 12}$, is a bit more complex because its denominator is a quadratic expression.

Why is this important? Recognizing the structure of the expression is the first step to solving it. We see that we're dealing with subtraction of fractions, which means we'll eventually need a common denominator. But before we get there, we need to simplify the expressions as much as possible. This involves factoring the quadratic expression in the denominator of the second fraction.

Factoring is Key: The denominator $2x^2 - 11x + 12$ is a quadratic expression, and factoring it will be crucial. Factoring helps us simplify the fraction and potentially find common factors with the denominator of the first fraction. This is like finding the right tool in your toolbox – factoring is the key to unlocking the simplification process.

Why Factoring Matters: Factoring not only simplifies the expression but also helps in identifying any restrictions on the variable x. Remember, the denominator of a fraction cannot be zero. So, by factoring, we can determine the values of x that would make the denominator zero, and these values will be excluded from the solution set. This is a critical step in ensuring our solution is mathematically sound. When we factor, we are essentially rewriting the quadratic expression as a product of two binomials, making it easier to work with and identify common factors.

Step-by-Step Solution

Now, let's dive into the actual steps to solve this expression. We'll break it down into manageable parts, making sure each step is clear and easy to follow.

1. Factoring the Quadratic Denominator

The first step is to factor the quadratic expression $2x^2 - 11x + 12$. We need to find two binomials that multiply to give us this quadratic. There are several techniques for factoring quadratics, such as the 'ac' method or trial and error. Let's use the 'ac' method.

• Identify a, b, and c: In our quadratic $2x^2 - 11x + 12$, a = 2, b = -11, and c = 12.
• Calculate ac: Multiply a and c: 2 * 12 = 24.
• Find factors of ac that add up to b: We need two numbers that multiply to 24 and add up to -11. These numbers are -8 and -3.
• Rewrite the middle term: Replace -11x with -8x - 3x. So, the quadratic becomes $2x^2 - 8x - 3x + 12$.
• Factor by grouping: Group the terms and factor out the greatest common factor (GCF) from each group:
  • $2x^2 - 8x$ becomes $2x(x - 4)$
  • $-3x + 12$ becomes $-3(x - 4)$
• Factor out the common binomial: Notice that both terms now have a common factor of (x - 4). Factor this out: $(x - 4)(2x - 3)$

So, the factored form of $2x^2 - 11x + 12$ is $(x - 4)(2x - 3)$. This is a crucial step because it simplifies the second fraction and reveals a common factor with the first fraction's denominator. By factoring, we've transformed a complex quadratic expression into a product of simpler binomials, making the subsequent steps much easier. This is like disassembling a complicated machine into its basic components – once we understand the individual parts, we can put them back together in a more efficient way.

2. Rewriting the Expression

Now that we've factored the denominator, we can rewrite the original expression as:

$\frac{5x}{x-4} - \frac{-2x+3}{(x-4)(2x-3)}$

Notice anything interesting? The denominator of the second fraction now has a factor of $(x - 4)$, which is the exact denominator of the first fraction. This is a big win! It means we're on the right track to finding a common denominator. Rewriting the expression in this form makes it crystal clear what our next step should be – to make the denominators the same. It's like having the pieces of a puzzle laid out in front of you – you can see how they fit together and what the next move should be.

The Importance of Observation: Taking a moment to observe the rewritten expression is key. We're not just blindly following steps; we're actively looking for patterns and simplifications. This is a critical skill in mathematics – to not just solve problems, but to understand the underlying structure and relationships. In this case, recognizing the common factor in the denominators is the key to efficiently solving the problem. It saves us time and effort in the long run.

3. Finding a Common Denominator

To subtract the fractions, we need a common denominator. In this case, the least common denominator (LCD) is $(x - 4)(2x - 3)$. The first fraction, $\frac{5x}{x-4}$, needs to be multiplied by $\frac{2x-3}{2x-3}$ to get the common denominator.

So, we have:

$\frac{5x}{x-4} \cdot \frac{2x-3}{2x-3} = \frac{5x(2x-3)}{(x-4)(2x-3)} = \frac{10x^2 - 15x}{(x-4)(2x-3)}$

Now, both fractions have the same denominator:

$\frac{10x^2 - 15x}{(x-4)(2x-3)} - \frac{-2x+3}{(x-4)(2x-3)}$

Finding a common denominator is like speaking the same language. Once both fractions are expressed with the same denominator, we can easily combine them. This step is crucial for simplifying the expression and moving closer to the final solution. Without a common denominator, we wouldn't be able to perform the subtraction. It's like trying to add apples and oranges – we need to convert them to a common unit before we can add them together.

4. Combining the Fractions

Now that we have a common denominator, we can combine the fractions by subtracting the numerators:

$\frac{10x^2 - 15x - (-2x+3)}{(x-4)(2x-3)}$

Be careful with the negative sign! Remember to distribute it to both terms in the second numerator:

$\frac{10x^2 - 15x + 2x - 3}{(x-4)(2x-3)}$

Now, combine like terms in the numerator:

$\frac{10x^2 - 13x - 3}{(x-4)(2x-3)}$

Combining the fractions is like merging two streams into one river. We're bringing the two fractions together into a single, unified expression. This step requires careful attention to detail, especially when dealing with negative signs. It's easy to make a mistake if we're not mindful of the distribution of the negative sign. Think of it as carefully pouring liquids from two containers into one – we need to make sure we don't spill anything in the process.

5. Simplifying the Numerator (If Possible)

Now, let's see if we can simplify the numerator, $10x^2 - 13x - 3$. We'll try to factor it. This might seem like a repeat of step one, but sometimes the numerator can be factored further, leading to additional simplification.

• Identify a, b, and c: In our quadratic $10x^2 - 13x - 3$, a = 10, b = -13, and c = -3.
• Calculate ac: Multiply a and c: 10 * -3 = -30.
• Find factors of ac that add up to b: We need two numbers that multiply to -30 and add up to -13. These numbers are -15 and 2.
• Rewrite the middle term: Replace -13x with -15x + 2x. So, the quadratic becomes $10x^2 - 15x + 2x - 3$.
• Factor by grouping: Group the terms and factor out the GCF from each group:
  • $10x^2 - 15x$ becomes $5x(2x - 3)$
  • $2x - 3$ remains as $1(2x - 3)$
• Factor out the common binomial: Factor out the common factor of (2x - 3): $(2x - 3)(5x + 1)$

So, the factored form of $10x^2 - 13x - 3$ is $(2x - 3)(5x + 1)$. This is a crucial simplification! By factoring the numerator, we've uncovered a common factor with the denominator, which will allow us to further simplify the expression.

6. Final Simplification

Now, we can rewrite the expression with the factored numerator:

$\frac{(2x - 3)(5x + 1)}{(x-4)(2x-3)}$

Notice that we have a common factor of $(2x - 3)$ in both the numerator and the denominator. We can cancel this out:

$\frac{(5x + 1)}{(x-4)}$

And there you have it! The simplified expression is $\frac{5x + 1}{x - 4}$. This is the final answer, but we're not quite done yet. We need to consider the restrictions on x.

Cancellation is like trimming away excess baggage. By canceling the common factor, we're making the expression as lean and mean as possible. This is a hallmark of good mathematical practice – always strive for the simplest form. It's not just about getting the right answer; it's about expressing it in the most elegant and efficient way.

7. Identifying Restrictions

Remember, the denominator of a fraction cannot be zero. So, we need to find the values of x that would make the original denominators zero and exclude them from our solution.

The original denominators were $(x - 4)$ and $(2x^2 - 11x + 12)$, which we factored as $(x - 4)(2x - 3)$.

• Set $(x - 4) = 0$: This gives us $x = 4$.
• Set $(2x - 3) = 0$: This gives us $x = \frac{3}{2}$.

So, the restrictions are $x \neq 4$ and $x \neq \frac{3}{2}$. These are crucial pieces of information! They tell us the values of x that are not allowed, ensuring our solution is mathematically sound.

Restrictions are like warning signs on a road. They tell us where we can't go. In mathematics, restrictions are just as important as the solution itself. They define the boundaries within which our solution is valid. Ignoring restrictions can lead to incorrect or undefined results. So, always remember to identify and state the restrictions when simplifying rational expressions.

The Final Solution

Therefore, the simplified expression is $\frac{5x + 1}{x - 4}$, with the restrictions $x \neq 4$ and $x \neq \frac{3}{2}$.

Congratulations! We've successfully navigated this algebraic expression. We started with a complex-looking problem and, by breaking it down into manageable steps, we arrived at a simplified solution. This is the power of mathematics – to take the complex and make it understandable. Remember, guys, the key is to understand each step and why we're doing it. Math isn't just about memorizing formulas; it's about understanding the logic and reasoning behind them. So, keep practicing, keep exploring, and keep unraveling those mathematical mysteries!

Answering the Questions

Now, let's address the questions presented in the table format.

1. Table 1

Original Table:

5x	2x - 3
(x - 4)	(2x^2 - 11x + 12)

Rewritten as a Question:

Fill in the missing parts of the table related to the expression $\frac{5x}{x-4} - \frac{-2x+3}{2x^2 - 11x + 12}$.

This table seems to be setting up the initial components of the expression. The first row likely represents the numerators, and the third row represents the denominators. The missing parts would logically be the individual terms of the expression before any simplification.

Answer:

The table represents the two fractions in the expression. The first fraction has a numerator of 5x and a denominator of (x-4). The second fraction has a numerator of (-2x + 3) (or 2x-3 if we consider the subtraction) and a denominator of (2x^2 - 11x + 12). The table is essentially a visual representation of the original problem, breaking it down into its component parts. It helps to organize the information and see the structure of the expression more clearly.

2. Table 2

Original Table:

5x	-1
=	+
(x - 4)	(x - 4)(2x - 3)

Rewritten as a Question:

Explain the steps represented in the table to combine the fractions $\frac{5x}{x-4}$ and $\frac{-2x+3}{2x^2 - 11x + 12}$.

This table appears to be outlining the process of finding a common denominator and combining the fractions. The “+” suggests that the negative sign has been distributed, and the “-1” might be a placeholder for the numerator of the second fraction after some manipulation.

Answer:

This table shows the process of combining the two fractions. It highlights the need for a common denominator. The first column represents the original first fraction. The second column likely represents the transformation of the second fraction after factoring the denominator and distributing the negative sign. The “+” indicates that the subtraction has been converted to addition by distributing the negative sign. The -1 likely represents a simplified form of the numerator of the second fraction after considering the common denominator and the negative sign. The bottom row shows the denominators, with the second denominator being factored. This table is a concise way of illustrating the key steps involved in combining the fractions.

Conclusion

Simplifying algebraic expressions can seem daunting, but by breaking them down into manageable steps and understanding the underlying concepts, it becomes a much more approachable task. Remember to always look for opportunities to factor, find common denominators, combine like terms, and simplify the result. And most importantly, don't forget to identify any restrictions on the variables! By following these steps, you'll be well on your way to mastering algebraic manipulations. Keep practicing, and you'll become a math whiz in no time!