How do you solve #x-(3x)/(2)=(x+7)/(x-11)#?

Question

Genesis Allen · Accepted Answer

#x-(3x)/2 = (x+7)/(x-11)# Simplify the left side of the equation:
#-x/2 = (x+7)/(x-11)# Multiply both sides by #(-2)(x-11)#
#x^2-11x = -2x-14# Re-arrange terms to the left side
#x^2-9x +14 = 0# Factor
#(x-7)(x-2)=0# State solutions:
#x=7# or #x=2#

Eva Abramson · Answer

undefined To solve the equation x - (3x)/2 = (x + 7)/(x - 11), we can follow these steps:

1. Multiply both sides of the equation by the common denominator (2(x - 11)) to eliminate the fractions.
2. Simplify the equation by distributing and combining like terms.
3. Solve for x by isolating the variable on one side of the equation.
4. Check the solution by substituting it back into the original equation.

The detailed solution is as follows:

1. 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)

2. Distribute and combine like terms:
   2x^2 - 22x - 3x^2 + 33x = 2x + 14

Simplifying:
   -x^2 + 11x = 2x + 14

3. 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.

4. 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.