Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Factor $3$ out of $3 x + 3$ .

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Step 2.1.1

Factor $3$ out of $3 x$ .

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

Step 2.1.2

Factor $3$ out of $3$ .

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

Step 2.1.3

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

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

Step 2.2

To write $\frac{3}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.3

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

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

Step 2.4

Write each expression with a common denominator of $(x + 1) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Reorder the factors of $(x + 1) \cdot 2$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Multiply $3$ by $2$ .

$\frac{6 - (x + 1)}{2 (x + 1)} - \frac{1}{3 (x + 1)} = 0$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Apply the distributive property.

$\frac{6 - x - 1 \cdot 1}{2 (x + 1)} - \frac{1}{3 (x + 1)} = 0$

Step 2.7.2

Multiply $- 1$ by $1$ .

$\frac{6 - x - 1}{2 (x + 1)} - \frac{1}{3 (x + 1)} = 0$

Step 2.7.3

Subtract $1$ from $6$ .

$\frac{- x + 5}{2 (x + 1)} - \frac{1}{3 (x + 1)} = 0$

Step 2.8

To write $\frac{- x + 5}{2 (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- x + 5}{2 (x + 1)} \cdot \frac{3}{3} - \frac{1}{3 (x + 1)} = 0$

Step 2.9

To write $- \frac{1}{3 (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- x + 5}{2 (x + 1)} \cdot \frac{3}{3} - \frac{1}{3 (x + 1)} \cdot \frac{2}{2} = 0$

Step 2.10

Write each expression with a common denominator of $6 (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.10.1

Multiply $\frac{- x + 5}{2 (x + 1)}$ by $\frac{3}{3}$ .

$\frac{(- x + 5) \cdot 3}{2 (x + 1) \cdot 3} - \frac{1}{3 (x + 1)} \cdot \frac{2}{2} = 0$

Step 2.10.2

Multiply $3$ by $2$ .

$\frac{(- x + 5) \cdot 3}{6 (x + 1)} - \frac{1}{3 (x + 1)} \cdot \frac{2}{2} = 0$

Step 2.10.3

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

$\frac{(- x + 5) \cdot 3}{6 (x + 1)} - \frac{2}{3 (x + 1) \cdot 2} = 0$

Step 2.10.4

Multiply $2$ by $3$ .

$\frac{(- x + 5) \cdot 3}{6 (x + 1)} - \frac{2}{6 (x + 1)} = 0$

Step 2.11

Combine the numerators over the common denominator.

$\frac{(- x + 5) \cdot 3 - 2}{6 (x + 1)} = 0$

Step 2.12

Simplify the numerator.

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Step 2.12.1

Apply the distributive property.

$\frac{- x \cdot 3 + 5 \cdot 3 - 2}{6 (x + 1)} = 0$

Step 2.12.2

Multiply $3$ by $- 1$ .

$\frac{- 3 x + 5 \cdot 3 - 2}{6 (x + 1)} = 0$

Step 2.12.3

Multiply $5$ by $3$ .

$\frac{- 3 x + 15 - 2}{6 (x + 1)} = 0$

Step 2.12.4

Subtract $2$ from $15$ .

$\frac{- 3 x + 13}{6 (x + 1)} = 0$

Step 2.13

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

$\frac{- (3 x) + 13}{6 (x + 1)} = 0$

Step 2.14

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (3 x) - 1 (- 13)}{6 (x + 1)} = 0$

Step 2.15

Factor $- 1$ out of $- (3 x) - 1 (- 13)$ .

$\frac{- (3 x - 13)}{6 (x + 1)} = 0$

Step 2.16

Rewrite $- (3 x - 13)$ as $- 1 (3 x - 13)$ .

$\frac{- 1 (3 x - 13)}{6 (x + 1)} = 0$

Step 2.17

Move the negative in front of the fraction.

$- \frac{3 x - 13}{6 (x + 1)} = 0$

Step 3

Set the numerator equal to zero.

$3 x - 13 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $13$ to both sides of the equation.

$3 x = 13$

Step 4.2

Divide each term in $3 x = 13$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 x = 13$ by $3$ .

$\frac{3 x}{3} = \frac{13}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{13}{3}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{3}$