Fraction Fun: Spot The Error And Solve It!

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Hey math enthusiasts! Let's dive into a fun fraction problem that's got a tiny hiccup. We're going to break down the steps, find the mistake, and get to the correct solution. It's like a math detective story! This problem is all about adding fractions, a fundamental concept in algebra. Adding fractions involves finding a common denominator, which is the least common multiple (LCM) of the denominators of the fractions. Once you have the common denominator, you adjust the numerators accordingly and then add the numerators while keeping the common denominator. Let's see how our friend approached this problem. We need to identify the mistake in adding fractions and explain the steps to solve them correctly. By understanding the common mistakes and how to avoid them, you can boost your confidence in solving similar problems. This breakdown will not only help you understand the specific problem but also strengthen your overall understanding of fraction addition and algebraic manipulation.

Step-by-Step Breakdown

Let's take a look at the problem, step by step:

Step 1: $\frac{x}{(x-2)(x-3)} + \frac{3}{x+3}$

This is where we begin. We have two fractions that we want to add together. The first fraction has a denominator of (x-2)(x-3) and the second fraction has a denominator of (x+3). To add these fractions, we need to find a common denominator. The first step involves looking at the denominators of the fractions to be added. The denominators are (x-2)(x-3) and (x+3). To add these fractions, we need to find a common denominator. The least common denominator (LCD) will be the product of all the unique factors of the denominators. In this case, the LCD will be (x-2)(x-3)(x+3). Notice the expressions in the denominators. The goal is to get the same denominator for both fractions so that we can combine them easily. It's like comparing apples and oranges; you need to convert them to a common unit before you can add them. The common denominator here will be the product of all unique factors present in the individual denominators. So it will be a combination of (x-2), (x-3) and (x+3).

Step 2: $\frac{x}{(x-2)(x+3)} + \frac{3(x-2)}{(x-2)(x+3)}$

Here lies the error! The student incorrectly changed the first fraction's denominator from (x-2)(x-3) to (x-2)(x+3). And the second fraction's denominator should have (x-2)(x-3)(x+3) as a common denominator. Also, the original problem involved denominators (x-2)(x-3) and (x+3). To add these fractions, we need to get a common denominator. Let's think about this: the common denominator should be something that both (x-2)(x-3) and (x+3) can divide into evenly. A good choice would be the product of all the unique factors, so (x-2), (x-3), and (x+3). It looks like in step 2, the student messed up and tried to get a common denominator of (x-2)(x+3) for both fractions, and it is a wrong approach. When adding fractions, you must ensure that each fraction's denominator is adjusted appropriately to match the common denominator. Remember, whatever you do to the bottom (the denominator), you must also do to the top (the numerator) to keep the fraction's value the same. This step is about prepping our fractions so we can add them easily.

Step 3: $\frac{x+3x-6}{(x-2)(x+3)}$

This step is based on the incorrect step 2, and therefore is also incorrect. The numerators were combined based on the faulty common denominator from step 2. This step is a direct result of the previous incorrect step. You can see the numerator of the second fraction (from Step 2) 3(x-2) has been expanded and combined with the first fraction's numerator. The entire process of this step is built upon the initial mistake. Always check your work, especially when dealing with multiple steps. Sometimes a tiny error at the beginning can cause a domino effect. The objective here is to simplify the combined fraction, aiming to express the final result in its simplest form. Remember that simplifying involves combining like terms in the numerator and canceling out common factors if possible. At this stage, it's essential to review each calculation to verify that all terms are combined correctly.

Step 4: $\frac{4x-6}{(x-2)(x+3)}$

This is the final, incorrect result. Because the previous steps had errors, so the final result is also wrong. The numerator was simplified, but the expression is still incorrect due to the initial mistake in determining the common denominator and adjusting the fractions. In this stage, you're simplifying the fraction you’ve created. This could involve combining like terms in the numerator or canceling out common factors between the numerator and denominator. Double-check to see if the numerator can be factored any further, and if there are any common factors with the denominator to simplify the fraction. The goal is to get your answer in the simplest possible form, so take this time to review and ensure that all steps are accurate.

Identifying the Mistake

The mistake is in Step 2. The student incorrectly changed the denominators and didn't find the correct common denominator. The first fraction's denominator was changed from (x-2)(x-3) to (x-2)(x+3). Also, the original problem involved denominators (x-2)(x-3) and (x+3). To add these fractions, the student must find the common denominator and adjust the fractions. In order to add these fractions correctly, you must first find the common denominator which is the least common multiple of the existing denominators. This means you must find the LCD, which will be the product of all unique factors present in the original denominators. Once you have determined the correct common denominator, you must adjust the fractions by multiplying the numerator and denominator of each fraction by the factors missing in its original denominator to match the common denominator. When adding fractions, it's crucial to ensure that all fractions share the same denominator before combining them. Always remember to multiply the numerators and denominators by the same factor. To be more clear, the correct common denominator would be (x-2)(x-3)(x+3). So, the student made a mistake in identifying the common denominator. This is a crucial step in the process, and any mistake will lead to an incorrect result.

The Correct Solution

Let's fix this and solve it correctly!

  1. Find the Common Denominator: The correct common denominator (LCD) is (x-2)(x-3)(x+3).

  2. Adjust the Fractions: Rewrite each fraction with the common denominator.

    • For the first fraction $\fracx}{(x-2)(x-3)}$, we need to multiply the numerator and denominator by (x+3) $\frac{x(x+3){(x-2)(x-3)(x+3)}$
    • For the second fraction $\frac3}{x+3}$, we need to multiply the numerator and denominator by (x-2)(x-3) $\frac{3(x-2)(x-3){(x-2)(x-3)(x+3)}$

  3. Combine the Fractions: Now that we have a common denominator, we can add the numerators.

    • The expression becomes: $\frac{x(x+3) + 3(x-2)(x-3)}{(x-2)(x-3)(x+3)}$

  4. Simplify: Expand and simplify the numerator.

    • x(x+3)+3(x−2)(x−3)=x2+3x+3(x2−5x+6)=x2+3x+3x2−15x+18=4x2−12x+18x(x+3) + 3(x-2)(x-3) = x^2 + 3x + 3(x^2 - 5x + 6) = x^2 + 3x + 3x^2 - 15x + 18 = 4x^2 - 12x + 18

  5. Final Answer:

    • The simplified expression is $\frac{4x^2 - 12x + 18}{(x-2)(x-3)(x+3)}$

Conclusion

Great job sticking with this problem! Remember, fraction addition is a cornerstone of algebra. By carefully finding the common denominator and correctly adjusting the numerators, you'll be well on your way to mastering these problems. Always double-check your steps, especially when dealing with multiple operations. Keep practicing, and you'll become a fraction-adding pro in no time! Keep an eye out for these common mistakes and always go back and check your work. Good job, and happy solving!