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

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

Evaluate

$$-\frac{1}{2\left(2x-1\right)\left(6x-1\right)}$$

Short Solution Steps

Divide 1 by 1 to get 1.

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

Factor 4x^{2}-1.

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

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

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

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

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

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

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

Combine like terms in 4x^{2}+2x-2x-1-4x^{2}.

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

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

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

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

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

Do the multiplications in 6x-1+2\left(1-2x\right).

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

Combine like terms in 6x-1+2-4x.

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

Multiply \frac{-1}{\left(2x-1\right)\left(2x+1\right)} times \frac{2x+1}{2\left(6x-1\right)} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2x+1 in both numerator and denominator.

$$\frac{-1}{2\left(2x-1\right)\left(6x-1\right)}$$

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

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

Use the distributive property to multiply 4x-2 by 6x-1 and combine like terms.

$$\frac{-1}{24x^{2}-16x+2}$$

Expand

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

Short Solution Steps

Divide 1 by 1 to get 1.

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

Factor 4x^{2}-1.

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

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

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

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

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

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

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

Combine like terms in 4x^{2}+2x-2x-1-4x^{2}.

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

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

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

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

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

Do the multiplications in 6x-1+2\left(1-2x\right).

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

Combine like terms in 6x-1+2-4x.

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

Multiply \frac{-1}{\left(2x-1\right)\left(2x+1\right)} times \frac{2x+1}{2\left(6x-1\right)} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2x+1 in both numerator and denominator.

$$\frac{-1}{2\left(2x-1\right)\left(6x-1\right)}$$

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

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

Use the distributive property to multiply 4x-2 by 6x-1 and combine like terms.

$$\frac{-1}{24x^{2}-16x+2}$$