Question

All steps for: x/x-2 + x-1/x+1= -1

Answer 1

`\large\boxed{x=0\ \vee\ x=1}`

Explanation:

`Domain:\\\\x-2\\eq0\ \wedge\ x+1\\eq0\\\\x\\eq2\ \wedge\ x\\eq-1\\\\\boxed{D:\ x\in\mathbb{R}-\{-1,\ 2\}}\\\\=============================`

`(x)/(x-2)+(x-1)/(x+1)=-1\qquad\text{subtract}\ (x-1)/(x+1)\ \text{from both sides}\\\\(x)/(x-2)=-1-(x-1)/(x+1)\\\\(x)/(x-2)=(-(x+1))/(x+1)+(-(x-1))/(x+1)\\\\(x)/(x-2)=(-(x+1)-(x-1))/(x+1)\\\\(x)/(x-2)=(-x-1-x+1)/(x+1)\\\\(x)/(x-2)=(-2x)/(x+1)\qquad\text{cross multiply}`

`x(x+1)=-2x(x-2)\qquad\text{use the distributive property}\\\\(x)(x)+(x)(1)=(-2x)(x)+(-2x)(-2)\\\\x^2+x=-2x^2+4x\qquad\text{add}\ 2x^2\ \text{to both sides}\\\\3x^2+x=4x\qquad\text{subtract 4x from both sides}\\\\3x^2-3x=0\qquad\text{distributive}\\\\3x(x-1)=0\iff 3x=0\ \vee\ x-1=0\\\\x=0\in D\ \vee\ x=1\in D`

Answer 2

Explanation:

I'm taking this to mean

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

Multiply through by (x - 2)*(x + 1) to get rid of the denominator on the left.

x(x + 1) + (x - 1)(x - 2) = -1 * (x - 2)(x + 1)

Remove the brackets on the left and right.

Be careful about the right side. Do it in two steps (or three)

x^2 + x + x^2 - 3x + 2 = - (x^2 - 2x + x - 2)

2x^2 - 2x + 2 = - (x^2 - x - 2)

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

Bring the right side to the left

2x^2 - 2x + 2 + x^2 - x - 2 = 0

3x^2 - 3x = 0

Factor this

x*(3x - 3) =0

x = 0

3x - 3 = 0

Add 3 to both sides.

3x = 3

Divide by 3

x = 3/3

So either x = 0

or

x = 1

Just to confirm that that is correct, a graph is included which shows the x roots are 0 and 1