Solving 1/(x+4) - 1/(x-7) = 11/30 A Step-by-Step Guide

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Hey guys! Today, we're diving deep into a math problem that might seem a bit tricky at first glance, but trust me, we'll break it down step by step. We're going to solve the equation 1/(x+4) - 1/(x-7) = 11/30, where x is not equal to -4 or 7 (because we can't divide by zero, right?). This kind of problem often pops up in algebra, and mastering it will definitely boost your math skills. So, let's put on our thinking caps and get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have an equation with fractions, and our goal is to find the value(s) of x that make the equation true. The equation involves rational expressions, which are just fractions with polynomials in the numerator and denominator. The restriction x тЙа -4, 7 is crucial because if x were -4 or 7, one of the denominators would be zero, making the fraction undefined.

So, to recap, we need to find the values of x that satisfy the equation 1/(x+4) - 1/(x-7) = 11/30, keeping in mind that x cannot be -4 or 7. This type of problem is a classic example of solving rational equations, and it requires a systematic approach to avoid errors. We'll be using techniques like finding a common denominator, simplifying fractions, and solving quadratic equations. Stick with me, and you'll see how it all comes together!

Step-by-Step Solution

Okay, let's get down to business and solve this equation step-by-step. We'll break it into manageable chunks so it's super clear.

1. Finding a Common Denominator

The first thing we need to do is combine the fractions on the left side of the equation. To do this, we need a common denominator. Remember how we do this with regular fractions? It's the same idea here. The common denominator for (x+4) and (x-7) is simply their product: (x+4)(x-7).

So, we rewrite the fractions with this common denominator:

[1/(x+4)] * [(x-7)/(x-7)] - [1/(x-7)] * [(x+4)/(x+4)] = 11/30

This gives us:

(x-7) / [(x+4)(x-7)] - (x+4) / [(x+4)(x-7)] = 11/30

2. Combining the Fractions

Now that we have a common denominator, we can combine the fractions on the left side. This means subtracting the numerators:

[(x-7) - (x+4)] / [(x+4)(x-7)] = 11/30

Simplify the numerator:

(x - 7 - x - 4) / [(x+4)(x-7)] = 11/30

This simplifies to:

-11 / [(x+4)(x-7)] = 11/30

3. Cross-Multiplication

Now we have a single fraction on each side of the equation, so we can use cross-multiplication to get rid of the fractions. This means multiplying the numerator of the left side by the denominator of the right side, and vice versa:

-11 * 30 = 11 * (x+4)(x-7)

This simplifies to:

-330 = 11(x+4)(x-7)

4. Simplifying the Equation

We can simplify this equation further by dividing both sides by 11:

-30 = (x+4)(x-7)

Now, let's expand the right side of the equation:

-30 = x^2 - 7x + 4x - 28

Combine like terms:

-30 = x^2 - 3x - 28

5. Forming a Quadratic Equation

To solve for x, we need to rearrange the equation into a quadratic equation in the standard form (ax^2 + bx + c = 0). Add 30 to both sides:

0 = x^2 - 3x - 28 + 30

This simplifies to:

0 = x^2 - 3x + 2

6. Solving the Quadratic Equation

We now have a quadratic equation: x^2 - 3x + 2 = 0. There are a couple of ways to solve this: factoring, completing the square, or using the quadratic formula. In this case, factoring is the easiest method.

We need to find two numbers that multiply to 2 and add up to -3. Those numbers are -1 and -2. So, we can factor the quadratic equation as:

(x - 1)(x - 2) = 0

Now, set each factor equal to zero and solve for x:

x - 1 = 0 or x - 2 = 0

This gives us two solutions:

x = 1 or x = 2

7. Checking the Solutions

It's always a good idea to check our solutions to make sure they are valid and don't violate any restrictions. Remember, x cannot be -4 or 7.

Our solutions are x = 1 and x = 2. Neither of these values are -4 or 7, so they are both valid solutions.

8. The Final Answer

Therefore, the solutions to the equation 1/(x+4) - 1/(x-7) = 11/30 are x = 1 and x = 2.

Why This Method Works

You might be wondering, why does this whole process work? Let's break down the key ideas:

  • Common Denominator: Finding a common denominator allows us to combine fractions, which is essential for simplifying the equation. It's like speaking the same language тАУ we can't add or subtract fractions unless they have the same denominator.
  • Cross-Multiplication: Cross-multiplication is a shortcut for eliminating fractions in an equation. It's based on the principle that if a/b = c/d, then ad = bc. This makes the equation easier to work with.
  • Quadratic Equation: When we simplify the equation, we end up with a quadratic equation. Quadratic equations have a standard form (ax^2 + bx + c = 0) and can be solved using various methods, like factoring or the quadratic formula.
  • Factoring: Factoring is a technique for breaking down a quadratic expression into two linear expressions. This allows us to find the values of x that make the expression equal to zero.
  • Checking Solutions: Checking our solutions is crucial because sometimes we might get extraneous solutions тАУ values that satisfy the simplified equation but not the original equation. This can happen when we square both sides of an equation or perform other operations that introduce new solutions.

By understanding these key ideas, you can tackle similar problems with confidence. It's not just about memorizing steps; it's about grasping the underlying concepts.

Common Mistakes to Avoid

When solving equations like this, it's easy to make a few common mistakes. Let's go over some of them so you can avoid these pitfalls:

  • Forgetting the Negative Sign: When subtracting fractions, remember to distribute the negative sign to all terms in the numerator. For example, in our problem, we had (x-7) - (x+4). It's crucial to distribute the negative sign to both the x and the 4, resulting in x - 7 - x - 4.
  • Incorrectly Combining Like Terms: Be careful when combining like terms. Make sure you are adding or subtracting the coefficients correctly. For instance, -7x + 4x should be -3x, not -11x.
  • Dividing by Zero: Always remember the restrictions on the variable. In our case, x cannot be -4 or 7 because that would make the denominator zero. If you get a solution that violates these restrictions, you need to discard it.
  • Not Checking Solutions: As we discussed earlier, checking solutions is vital. Don't skip this step, or you might end up with extraneous solutions.
  • Rushing Through the Steps: Math problems require careful attention to detail. Avoid rushing and double-check your work at each step.

By being aware of these common mistakes, you can significantly improve your accuracy and problem-solving skills.

Practice Problems

Now that we've solved this problem together, let's test your understanding with a few practice problems. Solving practice problems is the best way to solidify your skills and build confidence.

  1. Solve: 1/(x-2) - 1/(x+3) = 5/6
  2. Solve: 2/(x+1) + 3/(x-1) = 5/2
  3. Solve: 1/(x+5) - 1/(x-2) = 7/10

Try solving these problems on your own. If you get stuck, review the steps we discussed in this guide. Remember, practice makes perfect!

Conclusion

Solving rational equations like 1/(x+4) - 1/(x-7) = 11/30 might seem challenging initially, but with a systematic approach and a bit of practice, you can master them. We've covered the key steps, from finding a common denominator to solving a quadratic equation. We've also discussed common mistakes to avoid and provided practice problems to help you hone your skills.

Remember, the key to success in math is understanding the concepts, not just memorizing the steps. So, keep practicing, keep asking questions, and keep pushing yourself. You've got this!

If you have any questions or want to discuss other math topics, feel free to leave a comment below. Happy solving!