Question
Correct the error. \( \frac{x}{x+2}+\frac{4}{x-5}=\square \)
Ask by Dickson Savage. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The corrected expression is \( \frac{x^2 - x + 8}{(x + 2)(x - 5)} \).
Solution
To add
x/(x + 2) + 4/(x – 5)
you must use a common denominator. The common denominator is (x + 2)(x – 5). Multiply numerator and denominator of each fraction by the factor missing from its denominator:
• For x/(x + 2), multiply by (x – 5)/(x – 5).
• For 4/(x – 5), multiply by (x + 2)/(x + 2).
This gives
[x(x – 5)]/[(x + 2)(x – 5)] + [4(x + 2)]/[(x + 2)(x – 5)].
Now combine the fractions:
= [x(x – 5) + 4(x + 2)]/[(x + 2)(x – 5)].
Next, expand the numerators:
x(x – 5) = x² – 5x
4(x + 2) = 4x + 8
So the numerator becomes:
x² – 5x + 4x + 8 = x² – x + 8.
Thus, the correct expression is
(x² – x + 8)/[(x + 2)(x – 5)].
So, the error is corrected by writing
x/(x + 2) + 4/(x – 5) = (x² – x + 8)/[(x + 2)(x – 5)].
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
To solve the equation \( \frac{x}{x+2}+\frac{4}{x-5}=0 \), let's get a common denominator, which is \((x+2)(x-5)\). Rewriting the equation gives us: \[ \frac{x(x-5) + 4(x+2)}{(x+2)(x-5)} = 0 \] For the overall fraction to equal zero, the numerator must be zero. So, you simplify and set the numerator equal to zero: \[ x(x-5) + 4(x+2) = 0 \] Solving this will lead you to find the values of \(x\) that satisfy the equation!
