How do you solve #x(3x)/(2)=(x+7)/(x11)#?
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To solve the equation x  (3x)/2 = (x + 7)/(x  11), we can follow these steps:
 Multiply both sides of the equation by the common denominator (2(x  11)) to eliminate the fractions.
 Simplify the equation by distributing and combining like terms.
 Solve for x by isolating the variable on one side of the equation.
 Check the solution by substituting it back into the original equation.
The detailed solution is as follows:

Multiply both sides of the equation by 2(x  11): 2(x  11)(x)  2(x  11)(3x)/2 = 2(x  11)(x + 7)/(x  11)
Simplifying: 2x(x  11)  3x(x  11) = 2(x + 7)

Distribute and combine like terms: 2x^2  22x  3x^2 + 33x = 2x + 14
Simplifying: x^2 + 11x = 2x + 14

Rearrange the equation and solve for x: x^2 + 11x  2x  14 = 0
Simplifying: x^2 + 9x  14 = 0
Factoring or using the quadratic formula, we find: (x  2)(x  7) = 0
Therefore, x = 2 or x = 7.

Check the solutions by substituting them back into the original equation: For x = 2: 2  (3(2))/2 = (2 + 7)/(2  11) 2  3 = 9/9 1 = 1
For x = 7: 7  (3(7))/2 = (7 + 7)/(7  11) 7  21/2 = 14/4 7/2 = 7/2
Both solutions satisfy the original equation.
Therefore, the solutions to the equation x  (3x)/2 = (x + 7)/(x  11) are x = 2 and x = 7.
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