Solve 2x-1/2x-2x/2x-1-7/2x-4x^2

$$\frac { 2 x - 1 } { 2 x } - \frac { 2 x } { 2 x - 1 } - \frac { 7 } { 2 x - 4 x ^ { 2 } }$$

Evaluate

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

Short Solution Steps

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

$$\frac{\left(2x-1\right)\left(2x-1\right)}{2x\left(2x-1\right)}-\frac{2x\times 2x}{2x\left(2x-1\right)}-\frac{7}{2x-4x^{2}}$$

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

$$\frac{\left(2x-1\right)\left(2x-1\right)-2x\times 2x}{2x\left(2x-1\right)}-\frac{7}{2x-4x^{2}}$$

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

$$\frac{4x^{2}-2x-2x+1-4x^{2}}{2x\left(2x-1\right)}-\frac{7}{2x-4x^{2}}$$

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

$$\frac{-4x+1}{2x\left(2x-1\right)}-\frac{7}{2x-4x^{2}}$$

Factor 2x-4x^{2}.

$$\frac{-4x+1}{2x\left(2x-1\right)}-\frac{7}{2x\left(-2x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x\left(2x-1\right) and 2x\left(-2x+1\right) is 2x\left(2x-1\right). Multiply \frac{7}{2x\left(-2x+1\right)} times \frac{-1}{-1}.

$$\frac{-4x+1}{2x\left(2x-1\right)}-\frac{7\left(-1\right)}{2x\left(2x-1\right)}$$

Since \frac{-4x+1}{2x\left(2x-1\right)} and \frac{7\left(-1\right)}{2x\left(2x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x+1-7\left(-1\right)}{2x\left(2x-1\right)}$$

Do the multiplications in -4x+1-7\left(-1\right).

$$\frac{-4x+1+7}{2x\left(2x-1\right)}$$

Combine like terms in -4x+1+7.

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

Factor the expressions that are not already factored in \frac{-4x+8}{2x\left(2x-1\right)}.

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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

Expand

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

Short Solution Steps

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

$$\frac{\left(2x-1\right)\left(2x-1\right)}{2x\left(2x-1\right)}-\frac{2x\times 2x}{2x\left(2x-1\right)}-\frac{7}{2x-4x^{2}}$$

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

$$\frac{\left(2x-1\right)\left(2x-1\right)-2x\times 2x}{2x\left(2x-1\right)}-\frac{7}{2x-4x^{2}}$$

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

$$\frac{4x^{2}-2x-2x+1-4x^{2}}{2x\left(2x-1\right)}-\frac{7}{2x-4x^{2}}$$

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

$$\frac{-4x+1}{2x\left(2x-1\right)}-\frac{7}{2x-4x^{2}}$$

Factor 2x-4x^{2}.

$$\frac{-4x+1}{2x\left(2x-1\right)}-\frac{7}{2x\left(-2x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x\left(2x-1\right) and 2x\left(-2x+1\right) is 2x\left(2x-1\right). Multiply \frac{7}{2x\left(-2x+1\right)} times \frac{-1}{-1}.

$$\frac{-4x+1}{2x\left(2x-1\right)}-\frac{7\left(-1\right)}{2x\left(2x-1\right)}$$

Since \frac{-4x+1}{2x\left(2x-1\right)} and \frac{7\left(-1\right)}{2x\left(2x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x+1-7\left(-1\right)}{2x\left(2x-1\right)}$$

Do the multiplications in -4x+1-7\left(-1\right).

$$\frac{-4x+1+7}{2x\left(2x-1\right)}$$

Combine like terms in -4x+1+7.

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

Factor the expressions that are not already factored in \frac{-4x+8}{2x\left(2x-1\right)}.

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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