Find the Domain (6x)/((x^2-4)/((3x-9)/(2x+4)))

Problem

Domain of (6*x)/(x2−4)/(3*x−9)/(2*x+4)

Solution

1. Identify all denominators in the complex fraction. For the expression to be defined, no denominator can equal zero. The denominators are 2*x+4 (3*x−9)/(2*x+4) and (x2−4)/(3*x−9)/(2*x+4)

2. Set the first denominator to zero to find excluded values.

2*x+4=0

2*x=−4

x=−2

3. Set the numerator of the second denominator to zero, because if 3*x−9=0 the entire fraction (3*x−9)/(2*x+4) becomes zero, making the middle denominator zero.

3*x−9=0

3*x=9

x=3

4. Set the numerator of the third denominator to zero, because if x2−4=0 the entire fraction (x2−4)/… becomes zero, making the main denominator zero.

x2−4=0

x2=4

x=2

x=−2

5. Combine all excluded values. The values that make any part of the expression undefined are x=−2 x=2 and x=3

6. Write the domain in interval notation, excluding these specific points from the set of all real numbers.

Final Answer

Domain: *(−∞,−2)∪(−2,2)∪(2,3)∪(3,∞)

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