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

Question

Julia Adams · Accepted Answer

#x = 10/7 or x = -1#

The denominators of an equation containing fractions can be eliminated.

Find the LCM and multiply each term by the LCM of the denominators; the denominators can cancel; this is an alternative to finding a common denominator and converting all the numerators.

LCM = #color(red)(3(x+1)(x-1))#

#(color(red)(3(x+1)(x-1))xx2)/(x-1) -(2xxcolor(red)(3(x+1)(x-1)))/3 " " = (4xxcolor(red)(3(x+1)(x-1)))/(x+1)#

#(color(red)(3(x+1)cancel((x-1)))xx2)/cancel((x-1) ) -(2xxcolor(red)(cancel3(x+1)(x-1)))/cancel3 " "=4xxcolor(red)(3cancel((x+1))(x-1))/cancel((x+1))#

#3(x+1)xx2 -2(x+1)(x-1) = 4xx3(x+1)(x-1)#

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

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

#14x^2-6x-20 =0 " divide by 2"#

#7x^2 -3x -10 = 0#

#(7x - 10)(x+1) = 0" factorise"#

#if 7x -10 = 0" then "x = 10/7#

#if x+1 = 0" then "x = -1"#

Kennedy Adams · 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.