Question

11/(x - 7) + 4/(7 - x) = ?

• A. 15/(7 - x)
• B. 15/(x - 7)
• C. 7/(7 - x)
• D. 15
• E. (-7)/(7 - x)

Hint:

When denominators of fractions are of the form (a - b) and (b - a), remember that (b - a) = -(a - b). You can factor out a -1 from one of the denominators to easily find a common denominator.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem involves adding algebraic fractions. To add or subtract fractions, they must have a common denominator.
Step 2: Key Formula or Approach:
The key observation is that the denominators are negatives of each other: \( (7 - x) = -(x - 7) \). We can use this relationship to create a common denominator.
Step 3: Detailed Explanation:
The expression is: \[ \frac{11}{x - 7} + \frac{4}{7 - x} \] We can rewrite the second term's denominator: \[ \frac{4}{7 - x} = \frac{4}{-(x - 7)} = -\frac{4}{x - 7} \] Now substitute this back into the original expression: \[ \frac{11}{x - 7} - \frac{4}{x - 7} \] Since the fractions now have a common denominator, we can combine the numerators: \[ \frac{11 - 4}{x - 7} = \frac{7}{x - 7} \] This answer is correct, but it is not among the options in its current form. We need to see if it's equivalent to one of the options. Let's look at option (E): \(\frac{-7}{7 - x}\).
We can manipulate our answer to match this form: \[ \frac{7}{x - 7} = \frac{7}{- (7 - x)} = -\frac{7}{7 - x} = \frac{-7}{7 - x} \] This matches option (E).
Alternatively, we could have used \(7 - x\) as the common denominator from the start: \[ \frac{11}{x - 7} = \frac{11}{-(7 - x)} = -\frac{11}{7 - x} \] The original expression becomes: \[ -\frac{11}{7 - x} + \frac{4}{7 - x} = \frac{-11 + 4}{7 - x} = \frac{-7}{7 - x} \] This directly gives us the answer in the form of option (E).
Step 4: Final Answer:
The simplified expression is \(\frac{-7}{7 - x}\).