Precalculus Examples

$\frac{2 x}{x + 1} = \frac{2 x - 1}{x}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$2 x \cdot x = (x + 1) (2 x - 1)$

Step 2

Solve the equation for $x$ .

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Step 2.1

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

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Step 2.1.1

Move $x$ .

$2 (x \cdot x) = (x + 1) (2 x - 1)$

Step 2.1.2

Multiply $x$ by $x$ .

$2 x^{2} = (x + 1) (2 x - 1)$

Step 2.2

Simplify $(x + 1) (2 x - 1)$ .

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Step 2.2.1

Expand $(x + 1) (2 x - 1)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$2 x^{2} = x (2 x - 1) + 1 (2 x - 1)$

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.2.2.1.2

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

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Step 2.2.2.1.2.1

Move $x$ .

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

Step 2.2.2.1.2.2

Multiply $x$ by $x$ .

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

Step 2.2.2.1.3

Move $- 1$ to the left of $x$ .

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

Step 2.2.2.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.2.2.1.5

Multiply $2 x$ by $1$ .

$2 x^{2} = 2 x^{2} - x + 2 x + 1 \cdot - 1$

Step 2.2.2.1.6

Multiply $- 1$ by $1$ .

$2 x^{2} = 2 x^{2} - x + 2 x - 1$

Step 2.2.2.2

Add $- x$ and $2 x$ .

$2 x^{2} = 2 x^{2} + x - 1$

Step 2.3

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{2} + x - 1 = 2 x^{2}$

Step 2.4

Move all terms containing $x$ to the left side of the equation.

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Step 2.4.1

Subtract $2 x^{2}$ from both sides of the equation.

$2 x^{2} + x - 1 - 2 x^{2} = 0$

Step 2.4.2

Combine the opposite terms in $2 x^{2} + x - 1 - 2 x^{2}$ .

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Step 2.4.2.1

Subtract $2 x^{2}$ from $2 x^{2}$ .

$x - 1 + 0 = 0$

Step 2.4.2.2

Add $x - 1$ and $0$ .

$x - 1 = 0$

Step 2.5

Add $1$ to both sides of the equation.

$x = 1$