Algebra Examples

$\frac{2}{3 x} + \frac{4}{x} = \frac{7}{x + 1}$

Step 1

Subtract $\frac{7}{x + 1}$ from both sides of the equation.

$\frac{2}{3 x} + \frac{4}{x} - \frac{7}{x + 1} = 0$

Step 2

Simplify $\frac{2}{3 x} + \frac{4}{x} - \frac{7}{x + 1}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{2}{3 x}$ by $\frac{x + 1}{x + 1}$ .

$\frac{2}{3 x} \cdot \frac{x + 1}{x + 1} + \frac{4}{x} - \frac{7}{x + 1} = 0$

Step 2.1.2

Multiply $\frac{2}{3 x}$ by $\frac{x + 1}{x + 1}$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4}{x} - \frac{7}{x + 1} = 0$

Step 2.1.3

Multiply $\frac{4}{x}$ by $\frac{3 (x + 1)}{3 (x + 1)}$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4}{x} \cdot \frac{3 (x + 1)}{3 (x + 1)} - \frac{7}{x + 1} = 0$

Step 2.1.4

Multiply $\frac{4}{x}$ by $\frac{3 (x + 1)}{3 (x + 1)}$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4 (3 (x + 1))}{x (3 (x + 1))} - \frac{7}{x + 1} = 0$

Step 2.1.5

Multiply $\frac{7}{x + 1}$ by $\frac{3 x}{3 x}$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4 (3 (x + 1))}{x (3 (x + 1))} - (\frac{7}{x + 1} \cdot \frac{3 x}{3 x}) = 0$

Step 2.1.6

Multiply $\frac{7}{x + 1}$ by $\frac{3 x}{3 x}$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4 (3 (x + 1))}{x (3 (x + 1))} - \frac{7 (3 x)}{(x + 1) (3 x)} = 0$

Step 2.1.7

Reorder the factors of $x (3 (x + 1))$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4 (3 (x + 1))}{3 x (x + 1)} - \frac{7 (3 x)}{(x + 1) (3 x)} = 0$

Step 2.1.8

Reorder the factors of $(x + 1) (3 x)$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4 (3 (x + 1))}{3 x (x + 1)} - \frac{7 (3 x)}{3 x (x + 1)} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4 (3 (x + 1)) - 7 (3 x)}{3 x (x + 1)} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4 (3 x + 3 \cdot 1) - 7 (3 x)}{3 x (x + 1)} = 0$

Step 2.3.2

Multiply $3$ by $1$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4 (3 x + 3) - 7 (3 x)}{3 x (x + 1)} = 0$

Step 2.3.3

Apply the distributive property.

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{4 (3 x) + 4 \cdot 3 - 7 (3 x)}{3 x (x + 1)} = 0$

Step 2.3.4

Multiply $3$ by $4$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{12 x + 4 \cdot 3 - 7 (3 x)}{3 x (x + 1)} = 0$

Step 2.3.5

Multiply $4$ by $3$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{12 x + 12 - 7 (3 x)}{3 x (x + 1)} = 0$

Step 2.3.6

Multiply $3$ by $- 7$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{12 x + 12 - 21 x}{3 x (x + 1)} = 0$

Step 2.4

Subtract $21 x$ from $12 x$ .

$\frac{2 (x + 1)}{3 x (x + 1)} + \frac{- 9 x + 12}{3 x (x + 1)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{2 (x + 1) - 9 x + 12}{3 x (x + 1)} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

Apply the distributive property.

$\frac{2 x + 2 \cdot 1 - 9 x + 12}{3 x (x + 1)} = 0$

Step 2.6.1.2

Multiply $2$ by $1$ .

$\frac{2 x + 2 - 9 x + 12}{3 x (x + 1)} = 0$

Step 2.6.2

Subtract $9 x$ from $2 x$ .

$\frac{- 7 x + 2 + 12}{3 x (x + 1)} = 0$

Step 2.6.3

Add $2$ and $12$ .

$\frac{- 7 x + 14}{3 x (x + 1)} = 0$

Step 2.7

Factor $7$ out of $- 7 x + 14$ .

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Step 2.7.1

Factor $7$ out of $- 7 x$ .

$\frac{7 (- x) + 14}{3 x (x + 1)} = 0$

Step 2.7.2

Factor $7$ out of $14$ .

$\frac{7 (- x) + 7 (2)}{3 x (x + 1)} = 0$

Step 2.7.3

Factor $7$ out of $7 (- x) + 7 (2)$ .

$\frac{7 (- x + 2)}{3 x (x + 1)} = 0$

Step 2.8

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

$\frac{7 (- (x) + 2)}{3 x (x + 1)} = 0$

Step 2.9

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

$\frac{7 (- (x) - 1 (- 2))}{3 x (x + 1)} = 0$

Step 2.10

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

$\frac{7 (- (x - 2))}{3 x (x + 1)} = 0$

Step 2.11

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

$\frac{7 (- 1 (x - 2))}{3 x (x + 1)} = 0$

Step 2.12

Move the negative in front of the fraction.

$- \frac{7 (x - 2)}{3 x (x + 1)} = 0$

Step 3

Set the numerator equal to zero.

$7 (x - 2) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $7 (x - 2) = 0$ by $7$ and simplify.

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Step 4.1.1

Divide each term in $7 (x - 2) = 0$ by $7$ .

$\frac{7 (x - 2)}{7} = \frac{0}{7}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $7$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{7 (x - 2)}{7} = \frac{0}{7}$

Step 4.1.2.1.2

Divide $x - 2$ by $1$ .

$x - 2 = \frac{0}{7}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $7$ .

$x - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$x = 2$