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Chapter 0: Problem 7

Perform the indicated operations and simplify your answer. $$ \frac{5}{x-3}+\frac{3}{3-x} $$

Short Answer

Expert verified
The simplified form of the expression is \( \frac{2}{x-3} \).

Step by step solution

01

Identify the Common Denominator

To add these fractions, a common denominator is needed. However, the two denominators (x-3) and (3-x) look similar but are not identical. Thus, it is observed that (3-x) is the negative of (x-3). To ensure that the denominators are identical, the second fraction is multiplied by -1/-1 which does not change its value. Therefore, \(\frac{3}{3-x}\) becomes \(- \frac{3}{x-3}\).
02

Add the Fractions

Now that the fractions have a common denominator (x-3), they can be added together: \(\frac{5}{x-3} - \frac{3}{x-3}\). This is equal to \(\frac{5-3}{x-3}\).
03

Simplify the Fraction

After performing the subtraction in the numerator \(5-3 = 2\), the resulting fraction is \(\frac{2}{x-3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction addition
Adding fractions involves combining two or more fractions into a single fraction. However, before you can add them directly, you need to ensure they have the same denominator. This process makes it easier to manage the parts of the fractions. In the exercise you provided, fractions with denominators \( x-3 \) and \( 3-x \) seem different but can be manipulated to match, as one is simply the negative of the other.
  • Check the denominators for potential similarities, such as being negatives of each other.
  • Manipulate if necessary, which might involve multiplying by \( -1/-1 \) to make them equivalent.
  • Only with a common denominator can you proceed to the addition step.
Once the fractions share the same denominator, their numerators can be easily added or subtracted, resulting in a new single fraction.
Common denominator
Finding a common denominator is an essential part of adding fractions together. It simplifies the fractions so you can combine them more seamlessly. In our example, the two denominators, \( x-3 \) and \( 3-x \), look almost the same but differ in sign.To resolve this, recognize that \( 3-x \) is simply \( -(x-3) \). We can transform \( \frac{3}{3-x} \) into \(-\frac{3}{x-3}\) by multiplying both the numerator and denominator by \(-1\), a mathematical tactic that preserves the value of the original fraction but allows the denominators to align.
  • Identify any patterns or relationships in denominators, such as opposites.
  • Enable identical denominators by multiplying by \(-1/-1\) or other equivalent forms.
With aligned denominators, you can focus on adding the numerators and simplifying the result.
Simplification of fractions
Simplifying fractions is the final step to ensure the solution is presented in its simplest form. Once you have combined the fractions and obtained a single numerator and denominator, it's crucial to check if further simplification is possible.For the problem at hand, after adding the brought-to-same-denominator fractions, the solution becomes \( \frac{5}{x-3} - \frac{3}{x-3} = \frac{5-3}{x-3} \). You simplify the numerator by performing the subtraction: \( 5-3 = 2 \), giving you \( \frac{2}{x-3} \).
  • Look to simplify the numerator and denominator to the smallest possible integers.
  • Check if any factors can be cancelled or removed in both parts.
This ensures that the fraction is reduced to its simplest form, making it easier to work with and understand.

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Most popular questions from this chapter

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