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Fraction simplification is a key skill in algebra that often confuses students because it involves combining numbers in a fraction to make it easier to work with. It is essential to handle fractions correctly to avoid errors in your calculations. In this problem, the initial mistake was breaking the fraction by dividing each term separately instead of adding them first. To simplify a fraction correctly, always follow these steps:

• Add or subtract the numbers in the numerator if they appear in brackets.
• Multiply the numbers in the denominator unless instructed otherwise by the problem statement.
• Ensure to simplify the fraction as much as possible by finding the greatest common divisor.

For example, in the original exercise, the sum of 9 and 20 should have been treated as a single unit, giving us 29 in the numerator, with 15 as the denominator: \(\frac{29}{15}\). This correct form avoids misinterpretation and ensures accuracy in further calculations.Understanding how to properly simplify fractions helps in achieving accurate results, especially when combined with other operations like addition or subtraction.

Working with negative numbers can be tricky, but mastering them is essential as they frequently appear in algebra. These numbers differ from positive numbers in their position on the number line and how they interact with other numbers during arithmetic operations. When subtracting a negative number, it's crucial to remember that "minus a negative" is equivalent to "plus a positive". This transformation was applied in the exercise's step 3, where \(- (-3)\) becomes \(+ 3\). This understanding is fundamental to ensuring accurate addition and subtraction involving negatives:

• A negative times a negative results in a positive.
• A negative plus a negative increases the distance from zero in the negative direction.
• When combining numbers, visualize them on the number line for clarity.

Recognizing these properties of negative numbers helps solve mathematical problems correctly and simplifies complex expressions without errors.

Finding a common denominator is a crucial step when performing operations with fractions, especially addition and subtraction. The common denominator is a shared multiple of the denominators involved and allows fractions to be combined seamlessly.In the provided solution, after correctly simplifying fractions and handling the negative numbers, achieving a common denominator was necessary to add the fractions \(-9\), \(\frac{29}{15}\), and \(+3\). Here’s a quick guide on how to find a common denominator:

• Identify the denominators of all fractions involved.
• Find the least common multiple (LCM) of these numbers.
• Convert each fraction to an equivalent fraction using the LCM as the new denominator.
• Proceed with the arithmetic operation, be it addition or subtraction.

By applying these steps, the final answer could be reached as \(-61/15\), which ensured clarity and accuracy in the conversion to a mixed number with \(-4 \frac{1}{15}\). This process not only simplifies the arithmetic involved but also prepares the expressions for easier interpretation and presentation.