(x-2)+1/x+1=2/3

the way you typed it:

(x-2)+1/x+1=2/3
x-2 + 1/x + 1 = 2/3
x + 1/x -1 - 2/3 = 0
times 3x
3x^2 + 3 - 3x - 2x = 0
3x^2 - 5x + 3 = 0
x = (5 ± √-11)/6
no real solution

You probably meant:
x- 2 + 1/(x+1) = 2/3
times 3(x+1)
3x(x+1) - 6(x+1) + 3 = 2(x+1)
3x^2 + 3x - 6x - 6 + 3 = 2x + 2
3x^2 - 5x -5 = 0
x = (5 ±√85)/6

Why did you put brackets around x-2, which were not needed, and missed the ones where it counted ?

Heres my take....first combine like terms on the left side of equ......x-1/x+1 = 2/3

Now cross mult..... 3(x-1) = 2(x+1)

Distribute.....3x-3 = 2x +2
Subtr.....3x - 2x = 2 + 3
Yields....x = 5

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

Step 1: Simplify the equation if needed.
In this case, there is no immediate simplification we can do.

Step 2: Clear the fractions.
To eliminate the fractions, we multiply every term in the equation by the least common denominator (LCD) of all the fractions involved. The LCD in this case is 3(x+1) since it is the least common multiple of 3 and (x+1).

So, we multiply every term in the equation by 3(x+1):

3(x-2) + 3(1/x+1) = 3(2/3)

which simplifies to:

3(x-2) + 3(1)/(x+1) = 2

Step 3: Distribute the multiplication.
Distribute 3 to each term:

3x - 6 + 3/(x+1) = 2

Step 4: Combine like terms.
Combine the constant terms:

3x + 3/(x+1) - 6 = 2

Step 5: Get rid of the fraction.
To eliminate the fraction, we need to find a common denominator. In this case, the denominator is (x+1):

Multiply both sides of the equation by (x+1):

(x+1)(3x + 3/(x+1) - 6) = (x+1)(2)

Simplifying, we get:

3x(x+1) + 3 - 6(x+1) = 2x + 2

3x^2 + 3x + 3 - 6x - 6 = 2x + 2

Step 6: Simplify and solve the equation.
Combine like terms:

3x^2 - 3x - 3 = 2x + 2

Rearrange the equation to get all terms on one side:

3x^2 - 3x - 2x - 3 - 2 = 0

3x^2 - 5x - 5 = 0

Step 7: Solve the quadratic equation.
To solve the quadratic equation, we can use factoring, the quadratic formula, or completing the square. In this case, factoring cannot be easily done, so we will use the quadratic formula:

x = (-b ± √(b^2 - 4ac))/(2a)

In the quadratic equation 3x^2 - 5x - 5 = 0, a = 3, b = -5, and c = -5:

x = (-(-5) ± √((-5)^2 - 4*3*(-5)))/(2*3)

Simplifying further:

x = (5 ± √(25 + 60))/6

x = (5 ± √85)/6

So the solutions to the equation are:

x = (5 + √85)/6 and x = (5 - √85)/6