Algebra Examples

$\frac{x}{2 x + 2} = \frac{- 2 x}{4 x + 4} + \frac{2 x - 3}{x + 1}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{- 2 x}{4 x + 4}$ from both sides of the equation.

$\frac{x}{2 x + 2} - \frac{- 2 x}{4 x + 4} = \frac{2 x - 3}{x + 1}$

Step 1.2

Subtract $\frac{2 x - 3}{x + 1}$ from both sides of the equation.

$\frac{x}{2 x + 2} - \frac{- 2 x}{4 x + 4} - \frac{2 x - 3}{x + 1} = 0$

Step 2

Simplify $\frac{x}{2 x + 2} - \frac{- 2 x}{4 x + 4} - \frac{2 x - 3}{x + 1}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Factor $2$ out of $2 x + 2$ .

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Step 2.1.1.1

Factor $2$ out of $2 x$ .

$\frac{x}{2 (x) + 2} - \frac{- 2 x}{4 x + 4} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.1.2

Factor $2$ out of $2$ .

$\frac{x}{2 (x) + 2 (1)} - \frac{- 2 x}{4 x + 4} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.1.3

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

$\frac{x}{2 (x + 1)} - \frac{- 2 x}{4 x + 4} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.2

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

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Step 2.1.2.1

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

$\frac{x}{2 (x + 1)} - \frac{2 (- x)}{4 x + 4} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

Factor $2$ out of $4 x$ .

$\frac{x}{2 (x + 1)} - \frac{2 (- x)}{2 (2 x) + 4} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.2.2.2

Factor $2$ out of $4$ .

$\frac{x}{2 (x + 1)} - \frac{2 (- x)}{2 (2 x) + 2 (2)} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.2.2.3

Factor $2$ out of $2 (2 x) + 2 (2)$ .

$\frac{x}{2 (x + 1)} - \frac{2 (- x)}{2 (2 x + 2)} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.2.2.4

Cancel the common factor.

$\frac{x}{2 (x + 1)} - \frac{2 (- x)}{2 (2 x + 2)} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.2.2.5

Rewrite the expression.

$\frac{x}{2 (x + 1)} - \frac{- x}{2 x + 2} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.3

Factor $2$ out of $2 x + 2$ .

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Step 2.1.3.1

Factor $2$ out of $2 x$ .

$\frac{x}{2 (x + 1)} - \frac{- x}{2 (x) + 2} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.3.2

Factor $2$ out of $2$ .

$\frac{x}{2 (x + 1)} - \frac{- x}{2 (x) + 2 (1)} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.3.3

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

$\frac{x}{2 (x + 1)} - \frac{- x}{2 (x + 1)} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.4

Move the negative in front of the fraction.

$\frac{x}{2 (x + 1)} - - \frac{x}{2 (x + 1)} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.5

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

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Step 2.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{x}{2 (x + 1)} + 1 \frac{x}{2 (x + 1)} - \frac{2 x - 3}{x + 1} = 0$

Step 2.1.5.2

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

$\frac{x}{2 (x + 1)} + \frac{x}{2 (x + 1)} - \frac{2 x - 3}{x + 1} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{x + x}{2 (x + 1)} + \frac{- (2 x - 3)}{x + 1} = 0$

Step 2.3

Add $x$ and $x$ .

$\frac{2 x}{2 (x + 1)} + \frac{- (2 x - 3)}{x + 1} = 0$

Step 2.4

Simplify each term.

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Step 2.4.1

Cancel the common factor of $2$ .

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Step 2.4.1.1

Cancel the common factor.

$\frac{2 x}{2 (x + 1)} + \frac{- (2 x - 3)}{x + 1} = 0$

Step 2.4.1.2

Rewrite the expression.

$\frac{x}{x + 1} + \frac{- (2 x - 3)}{x + 1} = 0$

Step 2.4.2

Move the negative in front of the fraction.

$\frac{x}{x + 1} - \frac{2 x - 3}{x + 1} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{x - (2 x - 3)}{x + 1} = 0$

Step 2.6

Simplify each term.

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Step 2.6.1

Apply the distributive property.

$\frac{x - (2 x) - - 3}{x + 1} = 0$

Step 2.6.2

Multiply $2$ by $- 1$ .

$\frac{x - 2 x - - 3}{x + 1} = 0$

Step 2.6.3

Multiply $- 1$ by $- 3$ .

$\frac{x - 2 x + 3}{x + 1} = 0$

Step 2.7

Subtract $2 x$ from $x$ .

$\frac{- x + 3}{x + 1} = 0$

Step 2.8

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

$\frac{- (x) + 3}{x + 1} = 0$

Step 2.9

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- (x) - 1 (- 3)}{x + 1} = 0$

Step 2.10

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

$\frac{- (x - 3)}{x + 1} = 0$

Step 2.11

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$\frac{- 1 (x - 3)}{x + 1} = 0$

Step 2.12

Move the negative in front of the fraction.

$- \frac{x - 3}{x + 1} = 0$

Step 3

Set the numerator equal to zero.

$x - 3 = 0$

Step 4

Add $3$ to both sides of the equation.

$x = 3$