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

$$\frac { 2 t } { t + 2 } - \frac { 2 t ( 1 - t ) } { t + 2 } = - x + 3$$

Solve for x

$$x=\frac{6+3t-2t^{2}}{t+2}$$

$$t\neq -2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by t+2.

$$2t-2t\left(1-t\right)=-\left(t+2\right)x+\left(t+2\right)\times 3$$

Use the distributive property to multiply -2t by 1-t.

$$2t-2t+2t^{2}=-\left(t+2\right)x+\left(t+2\right)\times 3$$

Combine 2t and -2t to get 0.

$$2t^{2}=-\left(t+2\right)x+\left(t+2\right)\times 3$$

Use the distributive property to multiply t+2 by x.

$$2t^{2}=-\left(tx+2x\right)+\left(t+2\right)\times 3$$

To find the opposite of tx+2x, find the opposite of each term.

$$2t^{2}=-tx-2x+\left(t+2\right)\times 3$$

Use the distributive property to multiply t+2 by 3.

$$2t^{2}=-tx-2x+3t+6$$

Swap sides so that all variable terms are on the left hand side.

$$-tx-2x+3t+6=2t^{2}$$

Subtract 3t from both sides.

$$-tx-2x+6=2t^{2}-3t$$

Subtract 6 from both sides.

$$-tx-2x=2t^{2}-3t-6$$

Combine all terms containing x.

$$\left(-t-2\right)x=2t^{2}-3t-6$$

Divide both sides by -t-2.

$$\frac{\left(-t-2\right)x}{-t-2}=\frac{2t^{2}-3t-6}{-t-2}$$

Dividing by -t-2 undoes the multiplication by -t-2.

$$x=\frac{2t^{2}-3t-6}{-t-2}$$

Divide 2t^{2}-3t-6 by -t-2.

$$x=-\frac{2t^{2}-3t-6}{t+2}$$

Solve for t (complex solution)

$$t=\frac{\sqrt{\left(x-19\right)\left(x-3\right)}-x+3}{4}$$

$$t=\frac{-\sqrt{\left(x-19\right)\left(x-3\right)}-x+3}{4}$$

Solve for t

$$t=\frac{\sqrt{\left(x-19\right)\left(x-3\right)}-x+3}{4}$$

$$t=\frac{-\sqrt{\left(x-19\right)\left(x-3\right)}-x+3}{4}\text{, }x\leq 3\text{ or }x\geq 19$$