Algebra Examples

$\frac{5 x + 8}{- 3 x - 1}$

Step 1

Interchange the variables.

$x = \frac{5 y + 8}{- 3 y - 1}$

Step 2

Solve for $y$ .

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Step 2.1

Multiply the equation by $- 3 y - 1$ .

$(- 3 y - 1) (x) = (\frac{5 y + 8}{- 3 y - 1}) (- 3 y - 1)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $(- 3 y - 1) (x)$ .

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Step 2.2.1.1

Apply the distributive property.

$- 3 y x - 1 x = (\frac{5 y + 8}{- 3 y - 1}) (- 3 y - 1)$

Step 2.2.1.2

Rewrite $- 1 x$ as $- x$ .

$- 3 y x - x = (\frac{5 y + 8}{- 3 y - 1}) (- 3 y - 1)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify $(\frac{5 y + 8}{- 3 y - 1}) (- 3 y - 1)$ .

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Step 2.3.1.1

Multiply $\frac{5 y + 8}{- 3 y - 1}$ by $- 3 y - 1$ .

$- 3 y x - x = \frac{(5 y + 8) (- 3 y - 1)}{- 3 y - 1}$

Step 2.3.1.2

Cancel the common factor of $- 3 y - 1$ .

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Step 2.3.1.2.1

Cancel the common factor.

$- 3 y x - x = \frac{(5 y + 8) (- 3 y - 1)}{- 3 y - 1}$

Step 2.3.1.2.2

Divide $5 y + 8$ by $1$ .

$- 3 y x - x = 5 y + 8$

Step 2.4

Solve for $y$ .

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Step 2.4.1

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

$- 3 y x - x - 5 y = 8$

Step 2.4.2

Add $x$ to both sides of the equation.

$- 3 y x - 5 y = 8 + x$

Step 2.4.3

Factor $y$ out of $- 3 y x - 5 y$ .

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Step 2.4.3.1

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

$y (- 3 x) - 5 y = 8 + x$

Step 2.4.3.2

Factor $y$ out of $- 5 y$ .

$y (- 3 x) + y \cdot - 5 = 8 + x$

Step 2.4.3.3

Factor $y$ out of $y (- 3 x) + y \cdot - 5$ .

$y (- 3 x - 5) = 8 + x$

Step 2.4.4

Divide each term in $y (- 3 x - 5) = 8 + x$ by $- 3 x - 5$ and simplify.

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Step 2.4.4.1

Divide each term in $y (- 3 x - 5) = 8 + x$ by $- 3 x - 5$ .

$\frac{y (- 3 x - 5)}{- 3 x - 5} = \frac{8}{- 3 x - 5} + \frac{x}{- 3 x - 5}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $- 3 x - 5$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{y (- 3 x - 5)}{- 3 x - 5} = \frac{8}{- 3 x - 5} + \frac{x}{- 3 x - 5}$

Step 2.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{8}{- 3 x - 5} + \frac{x}{- 3 x - 5}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Combine the numerators over the common denominator.

$y = \frac{8 + x}{- 3 x - 5}$

Step 2.4.4.3.2

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

$y = \frac{8 + x}{- (3 x) - 5}$

Step 2.4.4.3.3

Rewrite $- 5$ as $- 1 (5)$ .

$y = \frac{8 + x}{- (3 x) - 1 (5)}$

Step 2.4.4.3.4

Factor $- 1$ out of $- (3 x) - 1 (5)$ .

$y = \frac{8 + x}{- (3 x + 5)}$

Step 2.4.4.3.5

Rewrite negatives.

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Step 2.4.4.3.5.1

Rewrite $- (3 x + 5)$ as $- 1 (3 x + 5)$ .

$y = \frac{8 + x}{- 1 (3 x + 5)}$

Step 2.4.4.3.5.2

Move the negative in front of the fraction.

$y = - \frac{8 + x}{3 x + 5}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = - \frac{8 + x}{3 x + 5}$

Step 4

Verify if $f^{- 1} (x) = - \frac{8 + x}{3 x + 5}$ is the inverse of $f (x) = \frac{5 x + 8}{- 3 x - 1}$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (\frac{5 x + 8}{- 3 x - 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{8 + \frac{5 x + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.3

Remove parentheses.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{8 + \frac{5 x + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.4

Simplify the numerator.

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Step 4.2.4.1

To write $8$ as a fraction with a common denominator, multiply by $\frac{- 3 x - 1}{- 3 x - 1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{\frac{8 (- 3 x - 1)}{- 3 x - 1} + \frac{5 x + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.4.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{\frac{8 (- 3 x - 1) + 5 x + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.4.3

Reorder terms.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{\frac{5 x + 8 (- 3 x - 1) + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.4.4

Rewrite $\frac{5 x + 8 (- 3 x - 1) + 8}{- 3 x - 1}$ in a factored form.

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Step 4.2.4.4.1

Apply the distributive property.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{\frac{5 x + 8 (- 3 x) + 8 \cdot - 1 + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.4.4.2

Multiply $- 3$ by $8$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{\frac{5 x - 24 x + 8 \cdot - 1 + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.4.4.3

Multiply $8$ by $- 1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{\frac{5 x - 24 x - 8 + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.4.4.4

Subtract $24 x$ from $5 x$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{\frac{- 19 x - 8 + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.4.4.5

Add $- 8$ and $8$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{\frac{- 19 x + 0}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.4.4.6

Add $- 19 x$ and $0$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{\frac{- 19 x}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.4.5

Move the negative in front of the fraction.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Step 4.2.5

Simplify the denominator.

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Step 4.2.5.1

Combine $3$ and $\frac{5 x + 8}{- 3 x - 1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{3 (5 x + 8)}{- 3 x - 1} + 5}$

Step 4.2.5.2

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{3 (5 x + 8)}{- 3 x - 1} + \frac{5 (- 3 x - 1)}{- 3 x - 1}}$

Step 4.2.5.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{3 (5 x + 8) + 5 (- 3 x - 1)}{- 3 x - 1}}$

Step 4.2.5.4

Reorder terms.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{5 (- 3 x - 1) + 3 (5 x + 8)}{- 3 x - 1}}$

Step 4.2.5.5

Rewrite $\frac{5 (- 3 x - 1) + 3 (5 x + 8)}{- 3 x - 1}$ in a factored form.

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Step 4.2.5.5.1

Apply the distributive property.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{5 (- 3 x) + 5 \cdot - 1 + 3 (5 x + 8)}{- 3 x - 1}}$

Step 4.2.5.5.2

Multiply $- 3$ by $5$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{- 15 x + 5 \cdot - 1 + 3 (5 x + 8)}{- 3 x - 1}}$

Step 4.2.5.5.3

Multiply $5$ by $- 1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{- 15 x - 5 + 3 (5 x + 8)}{- 3 x - 1}}$

Step 4.2.5.5.4

Apply the distributive property.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{- 15 x - 5 + 3 (5 x) + 3 \cdot 8}{- 3 x - 1}}$

Step 4.2.5.5.5

Multiply $5$ by $3$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{- 15 x - 5 + 15 x + 3 \cdot 8}{- 3 x - 1}}$

Step 4.2.5.5.6

Multiply $3$ by $8$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{- 15 x - 5 + 15 x + 24}{- 3 x - 1}}$

Step 4.2.5.5.7

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{0 - 5 + 24}{- 3 x - 1}}$

Step 4.2.5.5.8

Subtract $5$ from $0$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{- 5 + 24}{- 3 x - 1}}$

Step 4.2.5.5.9

Add $- 5$ and $24$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- \frac{19 x}{- 3 x - 1}}{\frac{19}{- 3 x - 1}}$

Step 4.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (- \frac{19 x}{- 3 x - 1} \cdot \frac{- 3 x - 1}{19})$

Step 4.2.7

Cancel the common factor of $19$ .

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Step 4.2.7.1

Move the leading negative in $- \frac{19 x}{- 3 x - 1}$ into the numerator.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{- 19 x}{- 3 x - 1} \cdot \frac{- 3 x - 1}{19})$

Step 4.2.7.2

Factor $19$ out of $- 19 x$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{19 (- x)}{- 3 x - 1} \cdot \frac{- 3 x - 1}{19})$

Step 4.2.7.3

Cancel the common factor.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{19 (- x)}{- 3 x - 1} \cdot \frac{- 3 x - 1}{19})$

Step 4.2.7.4

Rewrite the expression.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{- x}{- 3 x - 1} \cdot (- 3 x - 1))$

Step 4.2.8

Move the negative in front of the fraction.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (- \frac{x}{- 3 x - 1} \cdot (- 3 x - 1))$

Step 4.2.9

Apply the distributive property.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (- \frac{x}{- 3 x - 1} \cdot (- 3 x) - \frac{x}{- 3 x - 1} \cdot - 1)$

Step 4.2.10

Multiply $- \frac{x}{- 3 x - 1} (- 3 x)$ .

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Step 4.2.10.1

Multiply $- 3$ by $- 1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (3 (\frac{x}{- 3 x - 1}) \cdot x - \frac{x}{- 3 x - 1} \cdot - 1)$

Step 4.2.10.2

Combine $3$ and $\frac{x}{- 3 x - 1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x}{- 3 x - 1} \cdot x - \frac{x}{- 3 x - 1} \cdot - 1)$

Step 4.2.10.3

Combine $\frac{3 x}{- 3 x - 1}$ and $x$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x \cdot x}{- 3 x - 1} - \frac{x}{- 3 x - 1} \cdot - 1)$

Step 4.2.10.4

Raise $x$ to the power of $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 (x \cdot x)}{- 3 x - 1} - \frac{x}{- 3 x - 1} \cdot - 1)$

Step 4.2.10.5

Raise $x$ to the power of $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 (x \cdot x)}{- 3 x - 1} - \frac{x}{- 3 x - 1} \cdot - 1)$

Step 4.2.10.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x^{1 + 1}}{- 3 x - 1} - \frac{x}{- 3 x - 1} \cdot - 1)$

Step 4.2.10.7

Add $1$ and $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x^{2}}{- 3 x - 1} - \frac{x}{- 3 x - 1} \cdot - 1)$

Step 4.2.11

Multiply $- \frac{x}{- 3 x - 1} \cdot - 1$ .

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Step 4.2.11.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x^{2}}{- 3 x - 1} + 1 (\frac{x}{- 3 x - 1}))$

Step 4.2.11.2

Multiply $\frac{x}{- 3 x - 1}$ by $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x^{2}}{- 3 x - 1} + \frac{x}{- 3 x - 1})$

Step 4.2.12

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{3 x^{2} + x}{- 3 x - 1}$

Step 4.2.13

Factor $x$ out of $3 x^{2} + x$ .

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Step 4.2.13.1

Factor $x$ out of $3 x^{2}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (3 x) + x}{- 3 x - 1}$

Step 4.2.13.2

Raise $x$ to the power of $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (3 x) + x}{- 3 x - 1}$

Step 4.2.13.3

Factor $x$ out of $x^{1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (3 x) + x \cdot 1}{- 3 x - 1}$

Step 4.2.13.4

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (3 x + 1)}{- 3 x - 1}$

Step 4.2.14

Cancel the common factor of $3 x + 1$ and $- 3 x - 1$ .

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Step 4.2.14.1

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (- (- 3 x) + 1)}{- 3 x - 1}$

Step 4.2.14.2

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (- (- 3 x) - 1 \cdot - 1)}{- 3 x - 1}$

Step 4.2.14.3

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (- (- 3 x - 1))}{- 3 x - 1}$

Step 4.2.14.4

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (- 1 (- 3 x - 1))}{- 3 x - 1}$

Step 4.2.14.5

Cancel the common factor.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (- 1 (- 3 x - 1))}{- 3 x - 1}$

Step 4.2.14.6

Divide $x \cdot (- 1)$ by $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (x \cdot (- 1))$

Step 4.2.15

Simplify the expression.

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Step 4.2.15.1

Move $- 1$ to the left of $x$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (- 1 x)$

Step 4.2.15.2

Rewrite $- 1 x$ as $- x$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = x$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (- \frac{8 + x}{3 x + 5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{5 (- \frac{8 + x}{3 x + 5}) + 8}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

Multiply $5 (- \frac{8 + x}{3 x + 5})$ .

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Step 4.3.3.1.1

Multiply $- 1$ by $5$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{- 5 \frac{8 + x}{3 x + 5} + 8}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.1.2

Combine $- 5$ and $\frac{8 + x}{3 x + 5}$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{- 5 (8 + x)}{3 x + 5} + 8}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.2

Move the negative in front of the fraction.

$f (- \frac{8 + x}{3 x + 5}) = \frac{- \frac{5 (8 + x)}{3 x + 5} + 8}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{3 x + 5}{3 x + 5}$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{- \frac{5 (8 + x)}{3 x + 5} + \frac{8 (3 x + 5)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.4

Combine the numerators over the common denominator.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{- 5 (8 + x) + 8 (3 x + 5)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.5

Reorder terms.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{8 (3 x + 5) - 5 (8 + x)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.6

Rewrite $\frac{8 (3 x + 5) - 5 (8 + x)}{3 x + 5}$ in a factored form.

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Step 4.3.3.6.1

Apply the distributive property.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{8 (3 x) + 8 \cdot 5 - 5 (8 + x)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.6.2

Multiply $3$ by $8$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{24 x + 8 \cdot 5 - 5 (8 + x)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.6.3

Multiply $8$ by $5$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{24 x + 40 - 5 (8 + x)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.6.4

Apply the distributive property.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{24 x + 40 - 5 \cdot 8 - 5 x}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.6.5

Multiply $- 5$ by $8$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{24 x + 40 - 40 - 5 x}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.6.6

Subtract $5 x$ from $24 x$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x + 40 - 40}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.6.7

Subtract $40$ from $40$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x + 0}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.3.6.8

Add $19 x$ and $0$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Factor $- 1$ out of $- 3 (- \frac{8 + x}{3 x + 5}) - 1$ .

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Step 4.3.4.1.1

Reorder $8$ and $x$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- 3 \cdot (- 1 \frac{x + 8}{3 x + 5}) - 1}$

Step 4.3.4.1.2

Factor $- 1$ out of $- 3 \cdot - 1 \frac{x + 8}{3 x + 5}$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (- 3 \frac{x + 8}{3 x + 5}) - 1}$

Step 4.3.4.1.3

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

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (- 3 \frac{x + 8}{3 x + 5}) - 1 \cdot 1}$

Step 4.3.4.1.4

Factor $- 1$ out of $- (- 3 \frac{x + 8}{3 x + 5}) - 1 (1)$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (- 3 \frac{x + 8}{3 x + 5} + 1)}$

Step 4.3.4.2

Combine $- 3$ and $\frac{x + 8}{3 x + 5}$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (\frac{- 3 (x + 8)}{3 x + 5} + 1)}$

Step 4.3.4.3

Move the negative in front of the fraction.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (- \frac{3 (x + 8)}{3 x + 5} + 1)}$

Step 4.3.4.4

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

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (- \frac{3 (x + 8)}{3 x + 5} + \frac{3 x + 5}{3 x + 5})}$

Step 4.3.4.5

Combine the numerators over the common denominator.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{- 3 (x + 8) + 3 x + 5}{3 x + 5}}$

Step 4.3.4.6

Reorder terms.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{3 x + 5 - 3 (x + 8)}{3 x + 5}}$

Step 4.3.4.7

Rewrite $\frac{3 x + 5 - 3 (x + 8)}{3 x + 5}$ in a factored form.

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Step 4.3.4.7.1

Apply the distributive property.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{3 x + 5 - 3 x - 3 \cdot 8}{3 x + 5}}$

Step 4.3.4.7.2

Multiply $- 3$ by $8$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{3 x + 5 - 3 x - 24}{3 x + 5}}$

Step 4.3.4.7.3

Subtract $3 x$ from $3 x$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{0 + 5 - 24}{3 x + 5}}$

Step 4.3.4.7.4

Add $0$ and $5$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{5 - 24}{3 x + 5}}$

Step 4.3.4.7.5

Subtract $24$ from $5$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{- 19}{3 x + 5}}$

Step 4.3.4.8

Move the negative in front of the fraction.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{1 (\frac{19}{3 x + 5})}$

Step 4.3.4.9

Combine exponents.

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Step 4.3.4.9.1

Factor out negative.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{\frac{19}{3 x + 5}}$

Step 4.3.4.9.2

Multiply $- 1$ by $- 1$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{1 (\frac{19}{3 x + 5})}$

Step 4.3.4.9.3

Multiply $\frac{19}{3 x + 5}$ by $1$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{\frac{19}{3 x + 5}}$

Step 4.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{8 + x}{3 x + 5}) = \frac{19 x}{3 x + 5} \cdot \frac{3 x + 5}{19}$

Step 4.3.6

Cancel the common factor of $19$ .

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Step 4.3.6.1

Factor $19$ out of $19 x$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{19 (x)}{3 x + 5} \cdot \frac{3 x + 5}{19}$

Step 4.3.6.2

Cancel the common factor.

$f (- \frac{8 + x}{3 x + 5}) = \frac{19 x}{3 x + 5} \cdot \frac{3 x + 5}{19}$

Step 4.3.6.3

Rewrite the expression.

$f (- \frac{8 + x}{3 x + 5}) = \frac{x}{3 x + 5} \cdot (3 x + 5)$

Step 4.3.7

Cancel the common factor of $3 x + 5$ .

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Step 4.3.7.1

Cancel the common factor.

$f (- \frac{8 + x}{3 x + 5}) = \frac{x}{3 x + 5} \cdot (3 x + 5)$

Step 4.3.7.2

Rewrite the expression.

$f (- \frac{8 + x}{3 x + 5}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{8 + x}{3 x + 5}$ is the inverse of $f (x) = \frac{5 x + 8}{- 3 x - 1}$ .

$f^{- 1} (x) = - \frac{8 + x}{3 x + 5}$