Solve (1/1-4x^2/4x^2-1)+(1/2+1-2x/6x-1)=

$$( \frac { 1 } { 1 } - \frac { 4 x ^ { 2 } } { 4 x ^ { 2 } - 1 } ) + ( \frac { 1 } { 2 } + \frac { 1 - 2 x } { 6 x - 1 } ) =$$

Evaluate

$$\frac{8x^{3}+4x^{2}-14x+1}{2\left(6x-1\right)\left(4x^{2}-1\right)}$$

Short Solution Steps

Divide 1 by 1 to get 1.

$$1-\frac{4x^{2}}{4x^{2}-1}+\frac{1}{2}+\frac{1-2x}{6x-1}$$

Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$\frac{3}{2}-\frac{4x^{2}}{4x^{2}-1}+\frac{1-2x}{6x-1}$$

Factor 4x^{2}-1.

$$\frac{3}{2}-\frac{4x^{2}}{\left(2x-1\right)\left(2x+1\right)}+\frac{1-2x}{6x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and \left(2x-1\right)\left(2x+1\right) is 2\left(2x-1\right)\left(2x+1\right). Multiply \frac{3}{2} times \frac{\left(2x-1\right)\left(2x+1\right)}{\left(2x-1\right)\left(2x+1\right)}. Multiply \frac{4x^{2}}{\left(2x-1\right)\left(2x+1\right)} times \frac{2}{2}.

$$\frac{3\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(2x+1\right)}-\frac{2\times 4x^{2}}{2\left(2x-1\right)\left(2x+1\right)}+\frac{1-2x}{6x-1}$$

Since \frac{3\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(2x+1\right)} and \frac{2\times 4x^{2}}{2\left(2x-1\right)\left(2x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(2x-1\right)\left(2x+1\right)-2\times 4x^{2}}{2\left(2x-1\right)\left(2x+1\right)}+\frac{1-2x}{6x-1}$$

Do the multiplications in 3\left(2x-1\right)\left(2x+1\right)-2\times 4x^{2}.

$$\frac{12x^{2}+6x-6x-3-8x^{2}}{2\left(2x-1\right)\left(2x+1\right)}+\frac{1-2x}{6x-1}$$

Combine like terms in 12x^{2}+6x-6x-3-8x^{2}.

$$\frac{4x^{2}-3}{2\left(2x-1\right)\left(2x+1\right)}+\frac{1-2x}{6x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(2x-1\right)\left(2x+1\right) and 6x-1 is 2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right). Multiply \frac{4x^{2}-3}{2\left(2x-1\right)\left(2x+1\right)} times \frac{6x-1}{6x-1}. Multiply \frac{1-2x}{6x-1} times \frac{2\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(2x+1\right)}.

$$\frac{\left(4x^{2}-3\right)\left(6x-1\right)}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)}+\frac{\left(1-2x\right)\times 2\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)}$$

Since \frac{\left(4x^{2}-3\right)\left(6x-1\right)}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)} and \frac{\left(1-2x\right)\times 2\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4x^{2}-3\right)\left(6x-1\right)+\left(1-2x\right)\times 2\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)}$$

Do the multiplications in \left(4x^{2}-3\right)\left(6x-1\right)+\left(1-2x\right)\times 2\left(2x-1\right)\left(2x+1\right).

$$\frac{24x^{3}-4x^{2}-18x+3+8x^{2}-2-16x^{3}+4x}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)}$$

Combine like terms in 24x^{3}-4x^{2}-18x+3+8x^{2}-2-16x^{3}+4x.

$$\frac{8x^{3}+4x^{2}-14x+1}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)}$$

Expand 2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right).

$$\frac{8x^{3}+4x^{2}-14x+1}{48x^{3}-8x^{2}-12x+2}$$

Expand

$$\frac{8x^{3}+4x^{2}-14x+1}{2\left(6x-1\right)\left(4x^{2}-1\right)}$$

Short Solution Steps

Divide 1 by 1 to get 1.

$$1-\frac{4x^{2}}{4x^{2}-1}+\frac{1}{2}+\frac{1-2x}{6x-1}$$

Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$\frac{3}{2}-\frac{4x^{2}}{4x^{2}-1}+\frac{1-2x}{6x-1}$$

Factor 4x^{2}-1.

$$\frac{3}{2}-\frac{4x^{2}}{\left(2x-1\right)\left(2x+1\right)}+\frac{1-2x}{6x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and \left(2x-1\right)\left(2x+1\right) is 2\left(2x-1\right)\left(2x+1\right). Multiply \frac{3}{2} times \frac{\left(2x-1\right)\left(2x+1\right)}{\left(2x-1\right)\left(2x+1\right)}. Multiply \frac{4x^{2}}{\left(2x-1\right)\left(2x+1\right)} times \frac{2}{2}.

$$\frac{3\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(2x+1\right)}-\frac{2\times 4x^{2}}{2\left(2x-1\right)\left(2x+1\right)}+\frac{1-2x}{6x-1}$$

Since \frac{3\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(2x+1\right)} and \frac{2\times 4x^{2}}{2\left(2x-1\right)\left(2x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(2x-1\right)\left(2x+1\right)-2\times 4x^{2}}{2\left(2x-1\right)\left(2x+1\right)}+\frac{1-2x}{6x-1}$$

Do the multiplications in 3\left(2x-1\right)\left(2x+1\right)-2\times 4x^{2}.

$$\frac{12x^{2}+6x-6x-3-8x^{2}}{2\left(2x-1\right)\left(2x+1\right)}+\frac{1-2x}{6x-1}$$

Combine like terms in 12x^{2}+6x-6x-3-8x^{2}.

$$\frac{4x^{2}-3}{2\left(2x-1\right)\left(2x+1\right)}+\frac{1-2x}{6x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(2x-1\right)\left(2x+1\right) and 6x-1 is 2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right). Multiply \frac{4x^{2}-3}{2\left(2x-1\right)\left(2x+1\right)} times \frac{6x-1}{6x-1}. Multiply \frac{1-2x}{6x-1} times \frac{2\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(2x+1\right)}.

$$\frac{\left(4x^{2}-3\right)\left(6x-1\right)}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)}+\frac{\left(1-2x\right)\times 2\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)}$$

Since \frac{\left(4x^{2}-3\right)\left(6x-1\right)}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)} and \frac{\left(1-2x\right)\times 2\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4x^{2}-3\right)\left(6x-1\right)+\left(1-2x\right)\times 2\left(2x-1\right)\left(2x+1\right)}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)}$$

Do the multiplications in \left(4x^{2}-3\right)\left(6x-1\right)+\left(1-2x\right)\times 2\left(2x-1\right)\left(2x+1\right).

$$\frac{24x^{3}-4x^{2}-18x+3+8x^{2}-2-16x^{3}+4x}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)}$$

Combine like terms in 24x^{3}-4x^{2}-18x+3+8x^{2}-2-16x^{3}+4x.

$$\frac{8x^{3}+4x^{2}-14x+1}{2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right)}$$

Expand 2\left(2x-1\right)\left(6x-1\right)\left(2x+1\right).

$$\frac{8x^{3}+4x^{2}-14x+1}{48x^{3}-8x^{2}-12x+2}$$