Precalculus Examples

$\frac{11}{5 x^{3} - 45 x}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor the fraction.

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Step 1.1.1

Factor $5 x$ out of $5 x^{3} - 45 x$ .

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Step 1.1.1.1

Factor $5 x$ out of $5 x^{3}$ .

$\frac{11}{5 x (x^{2}) - 45 x}$

Step 1.1.1.2

Factor $5 x$ out of $- 45 x$ .

$\frac{11}{5 x (x^{2}) + 5 x (- 9)}$

Step 1.1.1.3

Factor $5 x$ out of $5 x (x^{2}) + 5 x (- 9)$ .

$\frac{11}{5 x (x^{2} - 9)}$

Step 1.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{11}{5 x (x^{2} - 3^{2})}$

Step 1.1.3

Factor.

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Step 1.1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{11}{5 x ((x + 3) (x - 3))}$

Step 1.1.3.2

Remove unnecessary parentheses.

$\frac{11}{5 x (x + 3) (x - 3)}$

Step 1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $B$ .

$\frac{A}{x} + \frac{B}{x + 3}$

Step 1.3

$\frac{A}{x} + \frac{B}{x + 3} + \frac{C}{x - 3}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $5 x (x + 3) (x - 3)$ .

$\frac{11 (5 x (x + 3) (x - 3))}{5 x (x + 3) (x - 3)} = \frac{A (5 x (x + 3) (x - 3))}{x} + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.5

Cancel the common factor of $5$ .

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Step 1.5.1

Cancel the common factor.

$\frac{11 (5 x (x + 3) (x - 3))}{5 x (x + 3) (x - 3)} = \frac{A (5 x (x + 3) (x - 3))}{x} + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.5.2

Rewrite the expression.

$\frac{11 ((x (x + 3)) (x - 3))}{(x (x + 3)) (x - 3)} = \frac{A (5 x (x + 3) (x - 3))}{x} + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.6

Cancel the common factor of $x$ .

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Step 1.6.1

Cancel the common factor.

$\frac{11 (x (x + 3) (x - 3))}{x (x + 3) (x - 3)} = \frac{A (5 x (x + 3) (x - 3))}{x} + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.6.2

Rewrite the expression.

$\frac{11 ((x + 3) (x - 3))}{(x + 3) (x - 3)} = \frac{A (5 x (x + 3) (x - 3))}{x} + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.7

Cancel the common factor of $x + 3$ .

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Step 1.7.1

Cancel the common factor.

$\frac{11 ((x + 3) (x - 3))}{(x + 3) (x - 3)} = \frac{A (5 x (x + 3) (x - 3))}{x} + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.7.2

Rewrite the expression.

$\frac{11 (x - 3)}{x - 3} = \frac{A (5 x (x + 3) (x - 3))}{x} + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.8

Cancel the common factor of $x - 3$ .

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Step 1.8.1

Cancel the common factor.

$\frac{11 (x - 3)}{x - 3} = \frac{A (5 x (x + 3) (x - 3))}{x} + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.8.2

Divide $11$ by $1$ .

$11 = \frac{A (5 x (x + 3) (x - 3))}{x} + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9

Simplify each term.

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Step 1.9.1

Cancel the common factor of $x$ .

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Step 1.9.1.1

Cancel the common factor.

$11 = \frac{A (5 x (x + 3) (x - 3))}{x} + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.1.2

Divide $A ((5 (x + 3)) (x - 3))$ by $1$ .

$11 = A ((5 (x + 3)) (x - 3)) + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.2

Rewrite using the commutative property of multiplication.

$11 = 5 A (x + 3) (x - 3) + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.3

Apply the distributive property.

$11 = (5 A x + 5 A \cdot 3) (x - 3) + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.4

Multiply $3$ by $5$ .

$11 = (5 A x + 15 A) (x - 3) + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.5

Expand $(5 A x + 15 A) (x - 3)$ using the FOIL Method.

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Step 1.9.5.1

Apply the distributive property.

$11 = 5 A x (x - 3) + 15 A (x - 3) + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.5.2

Apply the distributive property.

$11 = 5 A x \cdot x + 5 A x \cdot - 3 + 15 A (x - 3) + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.5.3

Apply the distributive property.

$11 = 5 A x \cdot x + 5 A x \cdot - 3 + 15 A x + 15 A \cdot - 3 + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.6

Simplify and combine like terms.

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Step 1.9.6.1

Simplify each term.

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Step 1.9.6.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.9.6.1.1.1

Move $x$ .

$11 = 5 A (x \cdot x) + 5 A x \cdot - 3 + 15 A x + 15 A \cdot - 3 + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.6.1.1.2

Multiply $x$ by $x$ .

$11 = 5 A x^{2} + 5 A x \cdot - 3 + 15 A x + 15 A \cdot - 3 + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.6.1.2

Multiply $- 3$ by $5$ .

$11 = 5 A x^{2} - 15 A x + 15 A x + 15 A \cdot - 3 + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.6.1.3

Multiply $- 3$ by $15$ .

$11 = 5 A x^{2} - 15 A x + 15 A x - 45 A + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.6.2

Add $- 15 A x$ and $15 A x$ .

$11 = 5 A x^{2} + 0 - 45 A + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.6.3

Add $5 A x^{2}$ and $0$ .

$11 = 5 A x^{2} - 45 A + \frac{(B) (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.7

Cancel the common factor of $x + 3$ .

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Step 1.9.7.1

Cancel the common factor.

$11 = 5 A x^{2} - 45 A + \frac{B (5 x (x + 3) (x - 3))}{x + 3} + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.7.2

Divide $(B) (5 x (x - 3))$ by $1$ .

$11 = 5 A x^{2} - 45 A + (B) (5 x (x - 3)) + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.8

Rewrite using the commutative property of multiplication.

$11 = 5 A x^{2} - 45 A + 5 B x (x - 3) + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.9

Apply the distributive property.

$11 = 5 A x^{2} - 45 A + 5 B x \cdot x + 5 B x \cdot - 3 + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.10

Multiply $x$ by $x$ by adding the exponents.

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Step 1.9.10.1

Move $x$ .

$11 = 5 A x^{2} - 45 A + 5 B (x \cdot x) + 5 B x \cdot - 3 + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.10.2

Multiply $x$ by $x$ .

$11 = 5 A x^{2} - 45 A + 5 B x^{2} + 5 B x \cdot - 3 + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.11

Multiply $- 3$ by $5$ .

$11 = 5 A x^{2} - 45 A + 5 B x^{2} - 15 B x + \frac{(C) (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.12

Cancel the common factor of $x - 3$ .

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Step 1.9.12.1

Cancel the common factor.

$11 = 5 A x^{2} - 45 A + 5 B x^{2} - 15 B x + \frac{C (5 x (x + 3) (x - 3))}{x - 3}$

Step 1.9.12.2

Divide $(C) (5 x (x + 3))$ by $1$ .

$11 = 5 A x^{2} - 45 A + 5 B x^{2} - 15 B x + (C) (5 x (x + 3))$

Step 1.9.13

Rewrite using the commutative property of multiplication.

$11 = 5 A x^{2} - 45 A + 5 B x^{2} - 15 B x + 5 C x (x + 3)$

Step 1.9.14

Apply the distributive property.

$11 = 5 A x^{2} - 45 A + 5 B x^{2} - 15 B x + 5 C x \cdot x + 5 C x \cdot 3$

Step 1.9.15

Multiply $x$ by $x$ by adding the exponents.

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Step 1.9.15.1

Move $x$ .

$11 = 5 A x^{2} - 45 A + 5 B x^{2} - 15 B x + 5 C (x \cdot x) + 5 C x \cdot 3$

Step 1.9.15.2

Multiply $x$ by $x$ .

$11 = 5 A x^{2} - 45 A + 5 B x^{2} - 15 B x + 5 C x^{2} + 5 C x \cdot 3$

Step 1.9.16

Multiply $3$ by $5$ .

$11 = 5 A x^{2} - 45 A + 5 B x^{2} - 15 B x + 5 C x^{2} + 15 C x$

Step 1.10

Reorder.

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Step 1.10.1

Move $A$ .

$11 = 5 x^{2} A - 45 A + 5 B x^{2} - 15 B x + 5 C x^{2} + 15 C x$

Step 1.10.2

Move $B$ .

$11 = 5 x^{2} A - 45 A + 5 x^{2} B - 15 B x + 5 C x^{2} + 15 C x$

Step 1.10.3

Move $B$ .

$11 = 5 x^{2} A - 45 A + 5 x^{2} B - 15 x B + 5 C x^{2} + 15 C x$

Step 1.10.4

Move $- 45 A$ .

$11 = 5 x^{2} A + 5 x^{2} B - 15 x B + 5 C x^{2} + 15 C x - 45 A$

Step 1.10.5

Move $- 15 x B$ .

$11 = 5 x^{2} A + 5 x^{2} B + 5 C x^{2} - 15 x B + 15 C x - 45 A$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = 5 A + 5 B + 5 C$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = - 15 B + 15 C$

Step 2.3

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$11 = - 45 A$

Step 2.4

Set up the system of equations to find the coefficients of the partial fractions.

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

$11 = - 45 A$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

$11 = - 45 A$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $11 = - 45 A$ .

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Step 3.1.1

Rewrite the equation as $- 45 A = 11$ .

$- 45 A = 11$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

Step 3.1.2

Divide each term in $- 45 A = 11$ by $- 45$ and simplify.

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Step 3.1.2.1

Divide each term in $- 45 A = 11$ by $- 45$ .

$\frac{- 45 A}{- 45} = \frac{11}{- 45}$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 45$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 45 A}{- 45} = \frac{11}{- 45}$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

Step 3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{11}{- 45}$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

$A = \frac{11}{- 45}$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

$A = \frac{11}{- 45}$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Move the negative in front of the fraction.

$A = - \frac{11}{45}$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

$A = - \frac{11}{45}$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

$A = - \frac{11}{45}$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

$A = - \frac{11}{45}$

$0 = 5 A + 5 B + 5 C$

$0 = - 15 B + 15 C$

Step 3.2

Replace all occurrences of $A$ with $- \frac{11}{45}$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $0 = 5 A + 5 B + 5 C$ with $- \frac{11}{45}$ .

$0 = 5 (- \frac{11}{45}) + 5 B + 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1.1

Move the leading negative in $- \frac{11}{45}$ into the numerator.

$0 = 5 (\frac{- 11}{45}) + 5 B + 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.2.2.1.1.2

Factor $5$ out of $45$ .

$0 = 5 (\frac{- 11}{5 (9)}) + 5 B + 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.2.2.1.1.3

Cancel the common factor.

$0 = 5 (\frac{- 11}{5 \cdot 9}) + 5 B + 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.2.2.1.1.4

Rewrite the expression.

$0 = \frac{- 11}{9} + 5 B + 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$0 = \frac{- 11}{9} + 5 B + 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.2.2.1.2

Move the negative in front of the fraction.

$0 = - \frac{11}{9} + 5 B + 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$0 = - \frac{11}{9} + 5 B + 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$0 = - \frac{11}{9} + 5 B + 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$0 = - \frac{11}{9} + 5 B + 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3

Solve for $B$ in $0 = - \frac{11}{9} + 5 B + 5 C$ .

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Step 3.3.1

Rewrite the equation as $- \frac{11}{9} + 5 B + 5 C = 0$ .

$- \frac{11}{9} + 5 B + 5 C = 0$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.2

Move all terms not containing $B$ to the right side of the equation.

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Step 3.3.2.1

Add $\frac{11}{9}$ to both sides of the equation.

$5 B + 5 C = \frac{11}{9}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.2.2

Subtract $5 C$ from both sides of the equation.

$5 B = \frac{11}{9} - 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$5 B = \frac{11}{9} - 5 C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3

Divide each term in $5 B = \frac{11}{9} - 5 C$ by $5$ and simplify.

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Step 3.3.3.1

Divide each term in $5 B = \frac{11}{9} - 5 C$ by $5$ .

$\frac{5 B}{5} = \frac{\frac{11}{9}}{5} + \frac{- 5 C}{5}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{5 B}{5} = \frac{\frac{11}{9}}{5} + \frac{- 5 C}{5}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{\frac{11}{9}}{5} + \frac{- 5 C}{5}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$B = \frac{\frac{11}{9}}{5} + \frac{- 5 C}{5}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$B = \frac{\frac{11}{9}}{5} + \frac{- 5 C}{5}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Simplify each term.

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Step 3.3.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$B = \frac{11}{9} \cdot \frac{1}{5} + \frac{- 5 C}{5}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3.3.1.2

Multiply $\frac{11}{9} \cdot \frac{1}{5}$ .

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Step 3.3.3.3.1.2.1

Multiply $\frac{11}{9}$ by $\frac{1}{5}$ .

$B = \frac{11}{9 \cdot 5} + \frac{- 5 C}{5}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3.3.1.2.2

Multiply $9$ by $5$ .

$B = \frac{11}{45} + \frac{- 5 C}{5}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$B = \frac{11}{45} + \frac{- 5 C}{5}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3.3.1.3

Cancel the common factor of $- 5$ and $5$ .

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Step 3.3.3.3.1.3.1

Factor $5$ out of $- 5 C$ .

$B = \frac{11}{45} + \frac{5 (- C)}{5}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3.3.1.3.2

Cancel the common factors.

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Step 3.3.3.3.1.3.2.1

Factor $5$ out of $5$ .

$B = \frac{11}{45} + \frac{5 (- C)}{5 (1)}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3.3.1.3.2.2

Cancel the common factor.

$B = \frac{11}{45} + \frac{5 (- C)}{5 \cdot 1}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3.3.1.3.2.3

Rewrite the expression.

$B = \frac{11}{45} + \frac{- C}{1}$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.3.3.3.1.3.2.4

Divide $- C$ by $1$ .

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - 15 B + 15 C$

Step 3.4

Replace all occurrences of $B$ with $\frac{11}{45} - C$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $0 = - 15 B + 15 C$ with $\frac{11}{45} - C$ .

$0 = - 15 (\frac{11}{45} - C) + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 15 (\frac{11}{45} - C) + 15 C$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$0 = - 15 (\frac{11}{45}) - 15 (- C) + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.4.2.1.1.2

Cancel the common factor of $15$ .

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Step 3.4.2.1.1.2.1

Factor $15$ out of $- 15$ .

$0 = 15 (- 1) (\frac{11}{45}) - 15 (- C) + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.4.2.1.1.2.2

Factor $15$ out of $45$ .

$0 = 15 \cdot (- 1 \frac{11}{15 \cdot 3}) - 15 (- C) + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.4.2.1.1.2.3

Cancel the common factor.

$0 = 15 \cdot (- 1 \frac{11}{15 \cdot 3}) - 15 (- C) + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.4.2.1.1.2.4

Rewrite the expression.

$0 = - 1 (\frac{11}{3}) - 15 (- C) + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - 1 (\frac{11}{3}) - 15 (- C) + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.4.2.1.1.3

Multiply $- 1$ by $- 15$ .

$0 = - 1 (\frac{11}{3}) + 15 C + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.4.2.1.1.4

Rewrite $- 1 (\frac{11}{3})$ as $- (\frac{11}{3})$ .

$0 = - \frac{11}{3} + 15 C + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - \frac{11}{3} + 15 C + 15 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.4.2.1.2

Add $15 C$ and $15 C$ .

$0 = - \frac{11}{3} + 30 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - \frac{11}{3} + 30 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - \frac{11}{3} + 30 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$0 = - \frac{11}{3} + 30 C$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.5

Solve for $C$ in $0 = - \frac{11}{3} + 30 C$ .

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Step 3.5.1

Rewrite the equation as $- \frac{11}{3} + 30 C = 0$ .

$- \frac{11}{3} + 30 C = 0$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.5.2

Add $\frac{11}{3}$ to both sides of the equation.

$30 C = \frac{11}{3}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.5.3

Divide each term in $30 C = \frac{11}{3}$ by $30$ and simplify.

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Step 3.5.3.1

Divide each term in $30 C = \frac{11}{3}$ by $30$ .

$\frac{30 C}{30} = \frac{\frac{11}{3}}{30}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $30$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{30 C}{30} = \frac{\frac{11}{3}}{30}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.5.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{\frac{11}{3}}{30}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$C = \frac{\frac{11}{3}}{30}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$C = \frac{\frac{11}{3}}{30}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$C = \frac{11}{3} \cdot \frac{1}{30}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.5.3.3.2

Multiply $\frac{11}{3} \cdot \frac{1}{30}$ .

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Step 3.5.3.3.2.1

Multiply $\frac{11}{3}$ by $\frac{1}{30}$ .

$C = \frac{11}{3 \cdot 30}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.5.3.3.2.2

Multiply $3$ by $30$ .

$C = \frac{11}{90}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$C = \frac{11}{90}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$C = \frac{11}{90}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$C = \frac{11}{90}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

$C = \frac{11}{90}$

$B = \frac{11}{45} - C$

$A = - \frac{11}{45}$

Step 3.6

Replace all occurrences of $C$ with $\frac{11}{90}$ in each equation.

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Step 3.6.1

Replace all occurrences of $C$ in $B = \frac{11}{45} - C$ with $\frac{11}{90}$ .

$B = \frac{11}{45} - (\frac{11}{90})$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $\frac{11}{45} - (\frac{11}{90})$ .

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Step 3.6.2.1.1

To write $\frac{11}{45}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$B = \frac{11}{45} \cdot \frac{2}{2} - \frac{11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

Step 3.6.2.1.2

Write each expression with a common denominator of $90$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.6.2.1.2.1

Multiply $\frac{11}{45}$ by $\frac{2}{2}$ .

$B = \frac{11 \cdot 2}{45 \cdot 2} - \frac{11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

Step 3.6.2.1.2.2

Multiply $45$ by $2$ .

$B = \frac{11 \cdot 2}{90} - \frac{11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

$B = \frac{11 \cdot 2}{90} - \frac{11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

Step 3.6.2.1.3

Combine the numerators over the common denominator.

$B = \frac{11 \cdot 2 - 11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

Step 3.6.2.1.4

Simplify the numerator.

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Step 3.6.2.1.4.1

Multiply $11$ by $2$ .

$B = \frac{22 - 11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

Step 3.6.2.1.4.2

Subtract $11$ from $22$ .

$B = \frac{11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

$B = \frac{11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

$B = \frac{11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

$B = \frac{11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

$B = \frac{11}{90}$

$C = \frac{11}{90}$

$A = - \frac{11}{45}$

Step 3.7

List all of the solutions.

$B = \frac{11}{90}, C = \frac{11}{90}, A = - \frac{11}{45}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x} + \frac{B}{x + 3} + \frac{C}{x - 3}$ with the values found for $A$ , $B$ , and $C$ .

$- \frac{\frac{11}{45}}{x} + \frac{\frac{11}{90}}{x + 3} + \frac{\frac{11}{90}}{x - 3}$