Solve 7*I+x+1/x-1/2-frac{x-1{x+1}}=2

$$7 \cdot \frac { I + \frac { x + 1 } { x - 1 } } { 2 - \frac { x - 1 } { x + 1 } } = 2$$

Solve for I (complex solution)

$$I=-\frac{5x^{2}+10x+13}{7\left(x^{2}-1\right)}$$

$$x\neq 1\text{ and }x\neq -3\text{ and }x\neq -1$$

Steps for Solving Linear Equation

Divide both sides by 7.

$$\frac{I+\frac{x+1}{x-1}}{2-\frac{x-1}{x+1}}=\frac{2}{7}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply I times \frac{x-1}{x-1}.

$$\frac{\frac{I\left(x-1\right)}{x-1}+\frac{x+1}{x-1}}{2-\frac{x-1}{x+1}}=\frac{2}{7}$$

Since \frac{I\left(x-1\right)}{x-1} and \frac{x+1}{x-1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{I\left(x-1\right)+x+1}{x-1}}{2-\frac{x-1}{x+1}}=\frac{2}{7}$$

Do the multiplications in I\left(x-1\right)+x+1.

$$\frac{\frac{Ix-I+x+1}{x-1}}{2-\frac{x-1}{x+1}}=\frac{2}{7}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x+1}{x+1}.

$$\frac{\frac{Ix-I+x+1}{x-1}}{\frac{2\left(x+1\right)}{x+1}-\frac{x-1}{x+1}}=\frac{2}{7}$$

Since \frac{2\left(x+1\right)}{x+1} and \frac{x-1}{x+1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{Ix-I+x+1}{x-1}}{\frac{2\left(x+1\right)-\left(x-1\right)}{x+1}}=\frac{2}{7}$$

Do the multiplications in 2\left(x+1\right)-\left(x-1\right).

$$\frac{\frac{Ix-I+x+1}{x-1}}{\frac{2x+2-x+1}{x+1}}=\frac{2}{7}$$

Combine like terms in 2x+2-x+1.

$$\frac{\frac{Ix-I+x+1}{x-1}}{\frac{x+3}{x+1}}=\frac{2}{7}$$

Divide \frac{Ix-I+x+1}{x-1} by \frac{x+3}{x+1} by multiplying \frac{Ix-I+x+1}{x-1} by the reciprocal of \frac{x+3}{x+1}.

$$\frac{\left(Ix-I+x+1\right)\left(x+1\right)}{\left(x-1\right)\left(x+3\right)}=\frac{2}{7}$$

Use the distributive property to multiply Ix-I+x+1 by x+1 and combine like terms.

$$\frac{Ix^{2}-I+x^{2}+2x+1}{\left(x-1\right)\left(x+3\right)}=\frac{2}{7}$$

Use the distributive property to multiply x-1 by x+3 and combine like terms.

$$\frac{Ix^{2}-I+x^{2}+2x+1}{x^{2}+2x-3}=\frac{2}{7}$$

Multiply both sides of the equation by 7\left(x-1\right)\left(x+3\right), the least common multiple of x^{2}+2x-3,7.

$$7\left(Ix^{2}-I+x^{2}+2x+1\right)=2\left(x-1\right)\left(x+3\right)$$

Use the distributive property to multiply 7 by Ix^{2}-I+x^{2}+2x+1.

$$7Ix^{2}-7I+7x^{2}+14x+7=2\left(x-1\right)\left(x+3\right)$$

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

$$7Ix^{2}-7I+7x^{2}+14x+7=\left(2x-2\right)\left(x+3\right)$$

Use the distributive property to multiply 2x-2 by x+3 and combine like terms.

$$7Ix^{2}-7I+7x^{2}+14x+7=2x^{2}+4x-6$$

Subtract 7x^{2} from both sides.

$$7Ix^{2}-7I+14x+7=2x^{2}+4x-6-7x^{2}$$

Combine 2x^{2} and -7x^{2} to get -5x^{2}.

$$7Ix^{2}-7I+14x+7=-5x^{2}+4x-6$$

Subtract 14x from both sides.

$$7Ix^{2}-7I+7=-5x^{2}+4x-6-14x$$

Combine 4x and -14x to get -10x.

$$7Ix^{2}-7I+7=-5x^{2}-10x-6$$

Subtract 7 from both sides.

$$7Ix^{2}-7I=-5x^{2}-10x-6-7$$

Subtract 7 from -6 to get -13.

$$7Ix^{2}-7I=-5x^{2}-10x-13$$

Combine all terms containing I.

$$\left(7x^{2}-7\right)I=-5x^{2}-10x-13$$

Divide both sides by 7x^{2}-7.

$$\frac{\left(7x^{2}-7\right)I}{7x^{2}-7}=\frac{-5x^{2}-10x-13}{7x^{2}-7}$$

Dividing by 7x^{2}-7 undoes the multiplication by 7x^{2}-7.

$$I=\frac{-5x^{2}-10x-13}{7x^{2}-7}$$

Divide -13-10x-5x^{2} by 7x^{2}-7.

$$I=-\frac{5x^{2}+10x+13}{7\left(x^{2}-1\right)}$$

Solve for I

$$I=-\frac{5x^{2}+10x+13}{7\left(x^{2}-1\right)}$$

$$x\neq -3\text{ and }|x|\neq 1$$

Steps for Solving Linear Equation

Divide both sides by 7.

$$\frac{I+\frac{x+1}{x-1}}{2-\frac{x-1}{x+1}}=\frac{2}{7}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply I times \frac{x-1}{x-1}.

$$\frac{\frac{I\left(x-1\right)}{x-1}+\frac{x+1}{x-1}}{2-\frac{x-1}{x+1}}=\frac{2}{7}$$

Since \frac{I\left(x-1\right)}{x-1} and \frac{x+1}{x-1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{I\left(x-1\right)+x+1}{x-1}}{2-\frac{x-1}{x+1}}=\frac{2}{7}$$

Do the multiplications in I\left(x-1\right)+x+1.

$$\frac{\frac{Ix-I+x+1}{x-1}}{2-\frac{x-1}{x+1}}=\frac{2}{7}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x+1}{x+1}.

$$\frac{\frac{Ix-I+x+1}{x-1}}{\frac{2\left(x+1\right)}{x+1}-\frac{x-1}{x+1}}=\frac{2}{7}$$

Since \frac{2\left(x+1\right)}{x+1} and \frac{x-1}{x+1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{Ix-I+x+1}{x-1}}{\frac{2\left(x+1\right)-\left(x-1\right)}{x+1}}=\frac{2}{7}$$

Do the multiplications in 2\left(x+1\right)-\left(x-1\right).

$$\frac{\frac{Ix-I+x+1}{x-1}}{\frac{2x+2-x+1}{x+1}}=\frac{2}{7}$$

Combine like terms in 2x+2-x+1.

$$\frac{\frac{Ix-I+x+1}{x-1}}{\frac{x+3}{x+1}}=\frac{2}{7}$$

Divide \frac{Ix-I+x+1}{x-1} by \frac{x+3}{x+1} by multiplying \frac{Ix-I+x+1}{x-1} by the reciprocal of \frac{x+3}{x+1}.

$$\frac{\left(Ix-I+x+1\right)\left(x+1\right)}{\left(x-1\right)\left(x+3\right)}=\frac{2}{7}$$

Use the distributive property to multiply Ix-I+x+1 by x+1 and combine like terms.

$$\frac{Ix^{2}-I+x^{2}+2x+1}{\left(x-1\right)\left(x+3\right)}=\frac{2}{7}$$

Use the distributive property to multiply x-1 by x+3 and combine like terms.

$$\frac{Ix^{2}-I+x^{2}+2x+1}{x^{2}+2x-3}=\frac{2}{7}$$

Multiply both sides of the equation by 7\left(x-1\right)\left(x+3\right), the least common multiple of x^{2}+2x-3,7.

$$7\left(Ix^{2}-I+x^{2}+2x+1\right)=2\left(x-1\right)\left(x+3\right)$$

Use the distributive property to multiply 7 by Ix^{2}-I+x^{2}+2x+1.

$$7Ix^{2}-7I+7x^{2}+14x+7=2\left(x-1\right)\left(x+3\right)$$

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

$$7Ix^{2}-7I+7x^{2}+14x+7=\left(2x-2\right)\left(x+3\right)$$

Use the distributive property to multiply 2x-2 by x+3 and combine like terms.

$$7Ix^{2}-7I+7x^{2}+14x+7=2x^{2}+4x-6$$

Subtract 7x^{2} from both sides.

$$7Ix^{2}-7I+14x+7=2x^{2}+4x-6-7x^{2}$$

Combine 2x^{2} and -7x^{2} to get -5x^{2}.

$$7Ix^{2}-7I+14x+7=-5x^{2}+4x-6$$

Subtract 14x from both sides.

$$7Ix^{2}-7I+7=-5x^{2}+4x-6-14x$$

Combine 4x and -14x to get -10x.

$$7Ix^{2}-7I+7=-5x^{2}-10x-6$$

Subtract 7 from both sides.

$$7Ix^{2}-7I=-5x^{2}-10x-6-7$$

Subtract 7 from -6 to get -13.

$$7Ix^{2}-7I=-5x^{2}-10x-13$$

Combine all terms containing I.

$$\left(7x^{2}-7\right)I=-5x^{2}-10x-13$$

Divide both sides by 7x^{2}-7.

$$\frac{\left(7x^{2}-7\right)I}{7x^{2}-7}=\frac{-5x^{2}-10x-13}{7x^{2}-7}$$

Dividing by 7x^{2}-7 undoes the multiplication by 7x^{2}-7.

$$I=\frac{-5x^{2}-10x-13}{7x^{2}-7}$$

Divide -5x^{2}-10x-13 by 7x^{2}-7.

$$I=-\frac{5x^{2}+10x+13}{7\left(x^{2}-1\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{49I^{2}-56I-40}+5}{7I+5}\text{, }&I\neq -\frac{5}{7}\\x=\frac{\sqrt{49I^{2}-56I-40}-5}{7I+5}\text{, }&I\neq -\frac{5}{7}\text{ and }I\neq -\frac{1}{2}\\x=-\frac{9}{5}\text{, }&I=-\frac{5}{7}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{49I^{2}-56I-40}+5}{7I+5}\text{, }&\left(I\neq -\frac{5}{7}\text{ and }I\leq \frac{4-2\sqrt{14}}{7}\right)\text{ or }I\geq \frac{2\sqrt{14}+4}{7}\\x=\frac{\sqrt{49I^{2}-56I-40}-5}{7I+5}\text{, }&I\geq \frac{2\sqrt{14}+4}{7}\text{ or }\left(I\neq -\frac{5}{7}\text{ and }I\leq \frac{4-2\sqrt{14}}{7}\text{ and }I\neq -\frac{1}{2}\right)\\x=-\frac{9}{5}\text{, }&I=-\frac{5}{7}\end{matrix}\right.$$