Algebra Examples

$\frac{2}{m} + \frac{3}{m + 9} = \frac{11}{4}$

Step 1

Subtract $\frac{11}{4}$ from both sides of the equation.

$\frac{2}{m} + \frac{3}{m + 9} - \frac{11}{4} = 0$

Step 2

Simplify $\frac{2}{m} + \frac{3}{m + 9} - \frac{11}{4}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{2}{m}$ by $\frac{(m + 9) \cdot 4}{(m + 9) \cdot 4}$ .

$\frac{2}{m} \cdot \frac{(m + 9) \cdot 4}{(m + 9) \cdot 4} + \frac{3}{m + 9} - \frac{11}{4} = 0$

Step 2.1.2

Multiply $\frac{2}{m}$ by $\frac{(m + 9) \cdot 4}{(m + 9) \cdot 4}$ .

$\frac{2 ((m + 9) \cdot 4)}{m ((m + 9) \cdot 4)} + \frac{3}{m + 9} - \frac{11}{4} = 0$

Step 2.1.3

Multiply $\frac{3}{m + 9}$ by $\frac{m \cdot 4}{m \cdot 4}$ .

$\frac{2 ((m + 9) \cdot 4)}{m ((m + 9) \cdot 4)} + \frac{3}{m + 9} \cdot \frac{m \cdot 4}{m \cdot 4} - \frac{11}{4} = 0$

Step 2.1.4

Multiply $\frac{3}{m + 9}$ by $\frac{m \cdot 4}{m \cdot 4}$ .

$\frac{2 ((m + 9) \cdot 4)}{m ((m + 9) \cdot 4)} + \frac{3 (m \cdot 4)}{(m + 9) (m \cdot 4)} - \frac{11}{4} = 0$

Step 2.1.5

Multiply $\frac{11}{4}$ by $\frac{m (m + 9)}{m (m + 9)}$ .

$\frac{2 ((m + 9) \cdot 4)}{m ((m + 9) \cdot 4)} + \frac{3 (m \cdot 4)}{(m + 9) (m \cdot 4)} - (\frac{11}{4} \cdot \frac{m (m + 9)}{m (m + 9)}) = 0$

Step 2.1.6

Multiply $\frac{11}{4}$ by $\frac{m (m + 9)}{m (m + 9)}$ .

$\frac{2 ((m + 9) \cdot 4)}{m ((m + 9) \cdot 4)} + \frac{3 (m \cdot 4)}{(m + 9) (m \cdot 4)} - \frac{11 (m (m + 9))}{4 (m (m + 9))} = 0$

Step 2.1.7

Reorder the factors of $m ((m + 9) \cdot 4)$ .

$\frac{2 ((m + 9) \cdot 4)}{4 m (m + 9)} + \frac{3 (m \cdot 4)}{(m + 9) (m \cdot 4)} - \frac{11 (m (m + 9))}{4 (m (m + 9))} = 0$

Step 2.1.8

Reorder the factors of $(m + 9) (m \cdot 4)$ .

$\frac{2 ((m + 9) \cdot 4)}{4 m (m + 9)} + \frac{3 (m \cdot 4)}{4 m (m + 9)} - \frac{11 (m (m + 9))}{4 (m (m + 9))} = 0$

Step 2.1.9

Reorder the factors of $4 (m (m + 9))$ .

$\frac{2 ((m + 9) \cdot 4)}{4 m (m + 9)} + \frac{3 (m \cdot 4)}{4 m (m + 9)} - \frac{11 (m (m + 9))}{4 m (m + 9)} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{2 ((m + 9) \cdot 4) + 3 (m \cdot 4) - 11 (m (m + 9))}{4 m (m + 9)} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{2 (m \cdot 4 + 9 \cdot 4) + 3 (m \cdot 4) - 11 (m (m + 9))}{4 m (m + 9)} = 0$

Step 2.3.2

Move $4$ to the left of $m$ .

$\frac{2 (4 \cdot m + 9 \cdot 4) + 3 (m \cdot 4) - 11 (m (m + 9))}{4 m (m + 9)} = 0$

Step 2.3.3

Multiply $9$ by $4$ .

$\frac{2 (4 \cdot m + 36) + 3 (m \cdot 4) - 11 (m (m + 9))}{4 m (m + 9)} = 0$

Step 2.3.4

Apply the distributive property.

$\frac{2 (4 m) + 2 \cdot 36 + 3 (m \cdot 4) - 11 (m (m + 9))}{4 m (m + 9)} = 0$

Step 2.3.5

Multiply $4$ by $2$ .

$\frac{8 m + 2 \cdot 36 + 3 (m \cdot 4) - 11 (m (m + 9))}{4 m (m + 9)} = 0$

Step 2.3.6

Multiply $2$ by $36$ .

$\frac{8 m + 72 + 3 (m \cdot 4) - 11 (m (m + 9))}{4 m (m + 9)} = 0$

Step 2.3.7

Move $4$ to the left of $m$ .

$\frac{8 m + 72 + 3 (4 \cdot m) - 11 (m (m + 9))}{4 m (m + 9)} = 0$

Step 2.3.8

Multiply $4$ by $3$ .

$\frac{8 m + 72 + 12 m - 11 (m (m + 9))}{4 m (m + 9)} = 0$

Step 2.3.9

Apply the distributive property.

$\frac{8 m + 72 + 12 m - 11 (m \cdot m + m \cdot 9)}{4 m (m + 9)} = 0$

Step 2.3.10

Multiply $m$ by $m$ .

$\frac{8 m + 72 + 12 m - 11 (m^{2} + m \cdot 9)}{4 m (m + 9)} = 0$

Step 2.3.11

Move $9$ to the left of $m$ .

$\frac{8 m + 72 + 12 m - 11 (m^{2} + 9 \cdot m)}{4 m (m + 9)} = 0$

Step 2.3.12

Apply the distributive property.

$\frac{8 m + 72 + 12 m - 11 m^{2} - 11 (9 m)}{4 m (m + 9)} = 0$

Step 2.3.13

Multiply $9$ by $- 11$ .

$\frac{8 m + 72 + 12 m - 11 m^{2} - 99 m}{4 m (m + 9)} = 0$

Step 2.4

Add $8 m$ and $12 m$ .

$\frac{20 m + 72 - 11 m^{2} - 99 m}{4 m (m + 9)} = 0$

Step 2.5

Subtract $99 m$ from $20 m$ .

$\frac{- 79 m + 72 - 11 m^{2}}{4 m (m + 9)} = 0$

Step 2.6

Factor by grouping.

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Step 2.6.1

Reorder terms.

$\frac{- 11 m^{2} - 79 m + 72}{4 m (m + 9)} = 0$

Step 2.6.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 11 \cdot 72 = - 792$ and whose sum is $b = - 79$ .

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Step 2.6.2.1

Factor $- 79$ out of $- 79 m$ .

$\frac{- 11 m^{2} - 79 (m) + 72}{4 m (m + 9)} = 0$

Step 2.6.2.2

Rewrite $- 79$ as $9$ plus $- 88$

$\frac{- 11 m^{2} + (9 - 88) m + 72}{4 m (m + 9)} = 0$

Step 2.6.2.3

Apply the distributive property.

$\frac{- 11 m^{2} + 9 m - 88 m + 72}{4 m (m + 9)} = 0$

Step 2.6.3

Factor out the greatest common factor from each group.

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Step 2.6.3.1

Group the first two terms and the last two terms.

$\frac{(- 11 m^{2} + 9 m) - 88 m + 72}{4 m (m + 9)} = 0$

Step 2.6.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{m (- 11 m + 9) + 8 (- 11 m + 9)}{4 m (m + 9)} = 0$

Step 2.6.4

Factor the polynomial by factoring out the greatest common factor, $- 11 m + 9$ .

$\frac{(- 11 m + 9) (m + 8)}{4 m (m + 9)} = 0$

Step 2.7

Factor $- 1$ out of $- 11 m$ .

$\frac{(- (11 m) + 9) (m + 8)}{4 m (m + 9)} = 0$

Step 2.8

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{(- (11 m) - 1 (- 9)) (m + 8)}{4 m (m + 9)} = 0$

Step 2.9

Factor $- 1$ out of $- (11 m) - 1 (- 9)$ .

$\frac{- (11 m - 9) (m + 8)}{4 m (m + 9)} = 0$

Step 2.10

Rewrite $- (11 m - 9)$ as $- 1 (11 m - 9)$ .

$\frac{- 1 (11 m - 9) (m + 8)}{4 m (m + 9)} = 0$

Step 2.11

Move the negative in front of the fraction.

$- \frac{(11 m - 9) (m + 8)}{4 m (m + 9)} = 0$

Step 3

Set the numerator equal to zero.

$(11 m - 9) (m + 8) = 0$

Step 4

Solve the equation for $m$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$11 m - 9 = 0$

$m + 8 = 0$

Step 4.2

Set $11 m - 9$ equal to $0$ and solve for $m$ .

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Step 4.2.1

Set $11 m - 9$ equal to $0$ .

$11 m - 9 = 0$

Step 4.2.2

Solve $11 m - 9 = 0$ for $m$ .

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Step 4.2.2.1

Add $9$ to both sides of the equation.

$11 m = 9$

Step 4.2.2.2

Divide each term in $11 m = 9$ by $11$ and simplify.

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Step 4.2.2.2.1

Divide each term in $11 m = 9$ by $11$ .

$\frac{11 m}{11} = \frac{9}{11}$

Step 4.2.2.2.2

Simplify the left side.

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Step 4.2.2.2.2.1

Cancel the common factor of $11$ .

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Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{11 m}{11} = \frac{9}{11}$

Step 4.2.2.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{9}{11}$

Step 4.3

Set $m + 8$ equal to $0$ and solve for $m$ .

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Step 4.3.1

Set $m + 8$ equal to $0$ .

$m + 8 = 0$

Step 4.3.2

Subtract $8$ from both sides of the equation.

$m = - 8$

Step 4.4

The final solution is all the values that make $(11 m - 9) (m + 8) = 0$ true.

$m = \frac{9}{11}, - 8$