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

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

Solve for a (complex solution)

$$a=\frac{1}{-4x-1}$$

$$x\neq 0\text{ and }x\neq -\frac{1}{4}\text{ and }x\neq -\frac{1}{2}$$

Steps for Solving Linear Equation

Variable a cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(a-1\right)\left(a+1\right), the least common multiple of a^{2}-1,a-1,a+1.

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

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

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

To find the opposite of 2ax+a+2x+1, find the opposite of each term.

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

Subtract 1 from 1 to get 0.

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

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

$$-2ax-a-2x=2ax-a-2x+1+a$$

Combine -a and a to get 0.

$$-2ax-a-2x=2ax-2x+1$$

Subtract 2ax from both sides.

$$-2ax-a-2x-2ax=-2x+1$$

Combine -2ax and -2ax to get -4ax.

$$-4ax-a-2x=-2x+1$$

Add 2x to both sides.

$$-4ax-a=-2x+1+2x$$

Combine -2x and 2x to get 0.

$$-4ax-a=1$$

Combine all terms containing a.

$$\left(-4x-1\right)a=1$$

Divide both sides by -4x-1.

$$\frac{\left(-4x-1\right)a}{-4x-1}=\frac{1}{-4x-1}$$

Dividing by -4x-1 undoes the multiplication by -4x-1.

$$a=\frac{1}{-4x-1}$$

Variable a cannot be equal to any of the values -1,1.

$$a=\frac{1}{-4x-1}\text{, }a\neq -1\text{ and }a\neq 1$$

Solve for x (complex solution)

$$x=-\frac{1}{4}-\frac{1}{4a}$$

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

Steps for Solving Linear Equation

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

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

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

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

To find the opposite of 2ax+a+2x+1, find the opposite of each term.

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

Subtract 1 from 1 to get 0.

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

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

$$-2ax-a-2x=2ax-a-2x+1+a$$

Combine -a and a to get 0.

$$-2ax-a-2x=2ax-2x+1$$

Subtract 2ax from both sides.

$$-2ax-a-2x-2ax=-2x+1$$

Combine -2ax and -2ax to get -4ax.

$$-4ax-a-2x=-2x+1$$

Add 2x to both sides.

$$-4ax-a-2x+2x=1$$

Combine -2x and 2x to get 0.

$$-4ax-a=1$$

Add a to both sides.

$$-4ax=1+a$$

The equation is in standard form.

$$\left(-4a\right)x=a+1$$

Divide both sides by -4a.

$$\frac{\left(-4a\right)x}{-4a}=\frac{a+1}{-4a}$$

Dividing by -4a undoes the multiplication by -4a.

$$x=\frac{a+1}{-4a}$$

Divide a+1 by -4a.

$$x=-\frac{1}{4}-\frac{1}{4a}$$

Solve for a

$$a=\frac{1}{-4x-1}$$

$$x\neq -\frac{1}{2}\text{ and }x\neq -\frac{1}{4}\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Variable a cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(a-1\right)\left(a+1\right), the least common multiple of a^{2}-1,a-1,a+1.

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

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

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

To find the opposite of 2ax+a+2x+1, find the opposite of each term.

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

Subtract 1 from 1 to get 0.

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

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

$$-2ax-a-2x=2ax-a-2x+1+a$$

Combine -a and a to get 0.

$$-2ax-a-2x=2ax-2x+1$$

Subtract 2ax from both sides.

$$-2ax-a-2x-2ax=-2x+1$$

Combine -2ax and -2ax to get -4ax.

$$-4ax-a-2x=-2x+1$$

Add 2x to both sides.

$$-4ax-a=-2x+1+2x$$

Combine -2x and 2x to get 0.

$$-4ax-a=1$$

Combine all terms containing a.

$$\left(-4x-1\right)a=1$$

Divide both sides by -4x-1.

$$\frac{\left(-4x-1\right)a}{-4x-1}=\frac{1}{-4x-1}$$

Dividing by -4x-1 undoes the multiplication by -4x-1.

$$a=\frac{1}{-4x-1}$$

Variable a cannot be equal to any of the values -1,1.

$$a=\frac{1}{-4x-1}\text{, }a\neq -1\text{ and }a\neq 1$$

Solve for x

$$x=-\frac{1}{4}-\frac{1}{4a}$$

$$a\neq 0\text{ and }|a|\neq 1$$

Steps for Solving Linear Equation

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

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

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

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

To find the opposite of 2ax+a+2x+1, find the opposite of each term.

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

Subtract 1 from 1 to get 0.

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

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

$$-2ax-a-2x=2ax-a-2x+1+a$$

Combine -a and a to get 0.

$$-2ax-a-2x=2ax-2x+1$$

Subtract 2ax from both sides.

$$-2ax-a-2x-2ax=-2x+1$$

Combine -2ax and -2ax to get -4ax.

$$-4ax-a-2x=-2x+1$$

Add 2x to both sides.

$$-4ax-a-2x+2x=1$$

Combine -2x and 2x to get 0.

$$-4ax-a=1$$

Add a to both sides.

$$-4ax=1+a$$

The equation is in standard form.

$$\left(-4a\right)x=a+1$$

Divide both sides by -4a.

$$\frac{\left(-4a\right)x}{-4a}=\frac{a+1}{-4a}$$

Dividing by -4a undoes the multiplication by -4a.

$$x=\frac{a+1}{-4a}$$

Divide a+1 by -4a.

$$x=-\frac{1}{4}-\frac{1}{4a}$$