Algebra Examples

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

Step 1

To write $\frac{2 x}{2 x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{2 x}{2 x + 1}$ by $\frac{x - 2}{x - 2}$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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Step 3.2.1

Move $x$ .

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

Step 3.2.2

Multiply $x$ by $x$ .

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

Step 3.3

Multiply $- 2$ by $2$ .

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Multiply $- 1$ by $- 1$ .

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

Step 3.6

Subtract $x$ from $- 4 x$ .

$\frac{2 x^{2} - 5 x + 1}{(2 x + 1) (x - 2)}$