Algebra Examples

$\frac{12 r}{r + 2} = \frac{4}{r + 2} - 6$

Step 1

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

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Step 1.1

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

$\frac{12 r}{r + 2} - \frac{4}{r + 2} = - 6$

Step 1.2

Add $6$ to both sides of the equation.

$\frac{12 r}{r + 2} - \frac{4}{r + 2} + 6 = 0$

Step 2

Simplify $\frac{12 r}{r + 2} - \frac{4}{r + 2} + 6$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Write $6$ as a fraction with denominator $1$ .

$\frac{12 r}{r + 2} - \frac{4}{r + 2} + \frac{6}{1} = 0$

Step 2.1.2

Multiply $\frac{6}{1}$ by $\frac{r + 2}{r + 2}$ .

$\frac{12 r}{r + 2} - \frac{4}{r + 2} + \frac{6}{1} \cdot \frac{r + 2}{r + 2} = 0$

Step 2.1.3

Multiply $\frac{6}{1}$ by $\frac{r + 2}{r + 2}$ .

$\frac{12 r}{r + 2} - \frac{4}{r + 2} + \frac{6 (r + 2)}{r + 2} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{12 r - 4 + 6 (r + 2)}{r + 2} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{12 r - 4 + 6 r + 6 \cdot 2}{r + 2} = 0$

Step 2.3.2

Multiply $6$ by $2$ .

$\frac{12 r - 4 + 6 r + 12}{r + 2} = 0$

Step 2.4

Add $12 r$ and $6 r$ .

$\frac{18 r - 4 + 12}{r + 2} = 0$

Step 2.5

Add $- 4$ and $12$ .

$\frac{18 r + 8}{r + 2} = 0$

Step 2.6

Factor $2$ out of $18 r + 8$ .

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Step 2.6.1

Factor $2$ out of $18 r$ .

$\frac{2 (9 r) + 8}{r + 2} = 0$

Step 2.6.2

Factor $2$ out of $8$ .

$\frac{2 (9 r) + 2 (4)}{r + 2} = 0$

Step 2.6.3

Factor $2$ out of $2 (9 r) + 2 (4)$ .

$\frac{2 (9 r + 4)}{r + 2} = 0$

Step 3

Set the numerator equal to zero.

$2 (9 r + 4) = 0$

Step 4

Solve the equation for $r$ .

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Step 4.1

Divide each term in $2 (9 r + 4) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (9 r + 4) = 0$ by $2$ .

$\frac{2 (9 r + 4)}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 (9 r + 4)}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $9 r + 4$ by $1$ .

$9 r + 4 = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$9 r + 4 = 0$

Step 4.2

Subtract $4$ from both sides of the equation.

$9 r = - 4$

Step 4.3

Divide each term in $9 r = - 4$ by $9$ and simplify.

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Step 4.3.1

Divide each term in $9 r = - 4$ by $9$ .

$\frac{9 r}{9} = \frac{- 4}{9}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $9$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{9 r}{9} = \frac{- 4}{9}$

Step 4.3.2.1.2

Divide $r$ by $1$ .

$r = \frac{- 4}{9}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Move the negative in front of the fraction.

$r = - \frac{4}{9}$