Solve ax-2/a+2+1+x/a-1=a-2x/a+2

$$\frac { a x - 2 } { a + 2 } + \frac { 1 + x } { a - 1 } = \frac { a - 2 x } { a + 2 }$$

Solve for x

$$x=\frac{a-2}{a}$$

$$a\neq 0\text{ and }a\neq 1\text{ and }a\neq -2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(a-1\right)\left(a+2\right), the least common multiple of a+2,a-1.

$$\left(a-1\right)\left(ax-2\right)+\left(a+2\right)\left(1+x\right)=\left(a-1\right)\left(a-2x\right)$$

Use the distributive property to multiply a-1 by ax-2.

$$xa^{2}-2a-ax+2+\left(a+2\right)\left(1+x\right)=\left(a-1\right)\left(a-2x\right)$$

Use the distributive property to multiply a+2 by 1+x.

$$xa^{2}-2a-ax+2+a+ax+2+2x=\left(a-1\right)\left(a-2x\right)$$

Combine -2a and a to get -a.

$$xa^{2}-a-ax+2+ax+2+2x=\left(a-1\right)\left(a-2x\right)$$

Combine -ax and ax to get 0.

$$xa^{2}-a+2+2+2x=\left(a-1\right)\left(a-2x\right)$$

Add 2 and 2 to get 4.

$$xa^{2}-a+4+2x=\left(a-1\right)\left(a-2x\right)$$

Use the distributive property to multiply a-1 by a-2x.

$$xa^{2}-a+4+2x=a^{2}-2ax-a+2x$$

Add 2ax to both sides.

$$xa^{2}-a+4+2x+2ax=a^{2}-a+2x$$

Subtract 2x from both sides.

$$xa^{2}-a+4+2x+2ax-2x=a^{2}-a$$

Combine 2x and -2x to get 0.

$$xa^{2}-a+4+2ax=a^{2}-a$$

Add a to both sides.

$$xa^{2}+4+2ax=a^{2}-a+a$$

Combine -a and a to get 0.

$$xa^{2}+4+2ax=a^{2}$$

Subtract 4 from both sides.

$$xa^{2}+2ax=a^{2}-4$$

Combine all terms containing x.

$$\left(a^{2}+2a\right)x=a^{2}-4$$

Divide both sides by a^{2}+2a.

$$\frac{\left(a^{2}+2a\right)x}{a^{2}+2a}=\frac{a^{2}-4}{a^{2}+2a}$$

Dividing by a^{2}+2a undoes the multiplication by a^{2}+2a.

$$x=\frac{a^{2}-4}{a^{2}+2a}$$

Divide a^{2}-4 by a^{2}+2a.

$$x=\frac{a-2}{a}$$

Solve for a (complex solution)

$$a=-\frac{2}{x-1}$$

$$x\neq 2\text{ and }x\neq 1\text{ and }x\neq -1$$

Solve for a

$$a=-\frac{2}{x-1}$$

$$x\neq 2\text{ and }|x|\neq 1$$