Simplify (2x+1-1/(1-2x))÷(2x-(4x^2)/(2x-1))

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

Step 1

Rewrite the division as a fraction.

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

Step 2

Multiply the numerator and denominator of the complex fraction by $(1 - 2 x) (2 x - 1)$ .

Multiply $\frac{2 x + 1 - \frac{1}{1 - 2 x}}{2 x - \frac{4 x^{2}}{2 x - 1}}$ by $\frac{(1 - 2 x) (2 x - 1)}{(1 - 2 x) (2 x - 1)}$ .

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

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

Cancel the common factor of $1 - 2 x$ .

Move the leading negative in $- \frac{1}{1 - 2 x}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2 x - 1$ .

Move the leading negative in $- \frac{4 x^{2}}{2 x - 1}$ into the numerator.

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

Factor $2 x - 1$ out of $(1 - 2 x) (2 x - 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Simplify the numerator.

Factor $2 x - 1$ out of $(1 - 2 x) (2 x - 1) (2 x) + (1 - 2 x) (2 x - 1) \cdot 1 + (2 x - 1) \cdot - 1$ .

Factor $2 x - 1$ out of $(1 - 2 x) (2 x - 1) (2 x)$ .

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

Factor $2 x - 1$ out of $(1 - 2 x) (2 x - 1) \cdot 1$ .

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

Factor $2 x - 1$ out of $(2 x - 1) \cdot - 1$ .

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

Factor $2 x - 1$ out of $(2 x - 1) ((1 - 2 x) (2 x)) + (2 x - 1) ((1 - 2 x) \cdot 1)$ .

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

Factor $2 x - 1$ out of $(2 x - 1) ((1 - 2 x) (2 x) + (1 - 2 x) \cdot 1) + (2 x - 1) (- 1)$ .

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

Apply the distributive property.

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

Multiply $2 x$ by $1$ .

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

Rewrite using the commutative property of multiplication.

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

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $2$ .

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

Multiply $1 - 2 x$ by $1$ .

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

Subtract $2 x$ from $2 x$ .

$\frac{(2 x - 1) (- 4 x^{2} + 1 + 0 - 1)}{(1 - 2 x) (2 x - 1) (2 x) + (1 - 2 x) (- 4 x^{2})}$

Add $- 4 x^{2}$ and $0$ .

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

Subtract $1$ from $1$ .

$\frac{(2 x - 1) (- 4 x^{2} + 0)}{(1 - 2 x) (2 x - 1) (2 x) + (1 - 2 x) (- 4 x^{2})}$

Add $- 4 x^{2}$ and $0$ .

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

Step 6

Simplify the denominator.

Factor $(1 - 2 x) \cdot 2 x$ out of $(1 - 2 x) (2 x - 1) (2 x) + (1 - 2 x) (- 4 x^{2})$ .

Factor $(1 - 2 x) \cdot 2 x$ out of $(1 - 2 x) (2 x - 1) (2 x)$ .

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

Factor $(1 - 2 x) \cdot 2 x$ out of $(1 - 2 x) (- 4 x^{2})$ .

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

Factor $(1 - 2 x) \cdot 2 x$ out of $(1 - 2 x) \cdot 2 x (2 x - 1) + (1 - 2 x) \cdot 2 x (- 2 x)$ .

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

Subtract $2 x$ from $2 x$ .

$\frac{(2 x - 1) \cdot - 4 x^{2}}{(1 - 2 x) \cdot 2 x (0 - 1)}$

Subtract $1$ from $0$ .

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

Multiply $- 1$ by $2$ .

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

Step 7

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $- 2$ .

Factor $- 2$ out of $(2 x - 1) \cdot - 4 x^{2}$ .

$\frac{- 2 ((2 x - 1) \cdot 2 x^{2})}{(1 - 2 x) \cdot - 2 x}$

Cancel the common factors.

Factor $- 2$ out of $(1 - 2 x) \cdot - 2 x$ .

$\frac{- 2 ((2 x - 1) \cdot 2 x^{2})}{- 2 ((1 - 2 x) x)}$

Cancel the common factor.

$\frac{- 2 ((2 x - 1) \cdot 2 x^{2})}{- 2 ((1 - 2 x) x)}$

Rewrite the expression.

$\frac{(2 x - 1) \cdot 2 x^{2}}{(1 - 2 x) x}$

Cancel the common factor of $x^{2}$ and $x$ .

Factor $x$ out of $(2 x - 1) \cdot 2 x^{2}$ .

$\frac{x ((2 x - 1) \cdot 2 x)}{(1 - 2 x) x}$

Cancel the common factors.

Factor $x$ out of $(1 - 2 x) x$ .

$\frac{x ((2 x - 1) \cdot 2 x)}{x (1 - 2 x)}$

Cancel the common factor.

$\frac{x ((2 x - 1) \cdot 2 x)}{x (1 - 2 x)}$

Rewrite the expression.

$\frac{(2 x - 1) \cdot 2 x}{1 - 2 x}$

Cancel the common factor of $2 x - 1$ and $1 - 2 x$ .

Factor $- 1$ out of $2 x$ .

$\frac{(- (- 2 x) - 1) \cdot 2 x}{1 - 2 x}$

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{(- (- 2 x) - 1 (1)) \cdot 2 x}{1 - 2 x}$

Factor $- 1$ out of $- (- 2 x) - 1 (1)$ .

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

Rewrite $- (- 2 x + 1)$ as $- 1 (- 2 x + 1)$ .

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

Reorder terms.

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

Cancel the common factor.

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

Divide $- 1 \cdot 2 x$ by $1$ .

$- 1 \cdot 2 x$

Multiply $- 1$ by $2$ .

$- 2 x$

Simplify (2x+1-1/(1-2x))÷(2x-(4x^2)/(2x-1))

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