How do you solve #2/(x-1) - 2/3 =4/(x+1)#?

Question

Abigail Allen · Accepted Answer

In an equation with fractions, we can get rid of the denominators by multiplying each term by the LCD (LCM of denominators).
IN this case the LCM is  #color(red)(3(x-1)(x+1))# #(color(red)(3(cancel(x-1))(x+1))xx2)/cancel(x-1) - (color(red)(cancel3(x-1)(x+1))xx2)/cancel3 =(color(red)(3(x-1)cancel(x+1))xx4)/(cancel(x+1)#   This results in a more straightforward equation after the denominators are cancelled. #6(x+1) -2color(blue)((x-1)(x+1))=12(x-1)"                   "color(blue)(DOTS)#       #6x+6 -2(x^2-1) = 12x-12# #6x+6 -2x^2+2=12x-12"                                   make =0"# #0 = 2x^2+6x-20"                                                           "div 2# #x^2+3x-10= 0"                                                         factorise"# #(x+5)(x-2)= 0# #x = -5 or x = 2#

Savannah Anderson · Answer

undefined To solve the equation 2/(x-1) - 2/3 = 4/(x+1), we can follow these steps:

1. Multiply every term in the equation by the least common denominator (LCD) of (x-1), 3, and (x+1), which is 3(x-1)(x+1). This step eliminates the denominators.

2. Simplify the equation by distributing and combining like terms.

3. Solve for x by isolating the variable on one side of the equation.

Here are the steps in detail:

1. Multiply every term by the LCD, 3(x-1)(x+1):

3(x-1)(x+1) * [2/(x-1) - 2/3] = 3(x-1)(x+1) * 4/(x+1)

Simplifying the left side: 3(x-1)(x+1) * 2/(x-1) - 3(x-1)(x+1) * 2/3

2. Distribute and combine like terms:

6(x+1) - 2(x-1)(x+1) = 12(x-1)

Simplifying further: 6x + 6 - 2(x^2 - 1) = 12x - 12

3. Simplify and solve for x:

6x + 6 - 2x^2 + 2 = 12x - 12

Rearranging the terms: -2x^2 + 6x + 8 = 12x - 12

Combining like terms: -2x^2 - 6x + 12x = -12 - 8

Simplifying further: -2x^2 + 6x = -20

Rearranging the terms: 2x^2 - 6x + 20 = 0

Factoring the quadratic equation is not possible, so we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values: x = (-(-6) ± √((-6)^2 - 4(2)(20))) / (2(2))

Simplifying: x = (6 ± √(36 - 160)) / 4

x = (6 ± √(-124)) / 4

Since the square root of a negative number is not a real number, this equation has no real solutions.