Algebra Examples

$- \frac{8 x}{5 x - 6}$

Step 1

Interchange the variables.

$x = - \frac{8 y}{5 y - 6}$

Step 2

Solve for $y$ .

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Step 2.1

Multiply the equation by $5 y - 6$ .

$(5 y - 6) (x) = (- \frac{8 y}{5 y - 6}) (5 y - 6)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Apply the distributive property.

$5 y x - 6 x = (- \frac{8 y}{5 y - 6}) (5 y - 6)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $5 y - 6$ .

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Step 2.3.1.1

Move the leading negative in $- \frac{8 y}{5 y - 6}$ into the numerator.

$5 y x - 6 x = \frac{- 8 y}{5 y - 6} (5 y - 6)$

Step 2.3.1.2

Cancel the common factor.

$5 y x - 6 x = \frac{- 8 y}{5 y - 6} (5 y - 6)$

Step 2.3.1.3

Rewrite the expression.

$5 y x - 6 x = - 8 y$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Add $8 y$ to both sides of the equation.

$5 y x - 6 x + 8 y = 0$

Step 2.4.2

Add $6 x$ to both sides of the equation.

$5 y x + 8 y = 6 x$

Step 2.4.3

Factor $y$ out of $5 y x + 8 y$ .

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Step 2.4.3.1

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

$y (5 x) + 8 y = 6 x$

Step 2.4.3.2

Factor $y$ out of $8 y$ .

$y (5 x) + y \cdot 8 = 6 x$

Step 2.4.3.3

Factor $y$ out of $y (5 x) + y \cdot 8$ .

$y (5 x + 8) = 6 x$

Step 2.4.4

Divide each term in $y (5 x + 8) = 6 x$ by $5 x + 8$ and simplify.

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Step 2.4.4.1

Divide each term in $y (5 x + 8) = 6 x$ by $5 x + 8$ .

$\frac{y (5 x + 8)}{5 x + 8} = \frac{6 x}{5 x + 8}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $5 x + 8$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{y (5 x + 8)}{5 x + 8} = \frac{6 x}{5 x + 8}$

Step 2.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{6 x}{5 x + 8}$

Step 3

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

$f^{- 1} (x) = \frac{6 x}{5 x + 8}$

Step 4

Verify if $f^{- 1} (x) = \frac{6 x}{5 x + 8}$ is the inverse of $f (x) = - \frac{8 x}{5 x - 6}$ .

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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 $f^{- 1} (- \frac{8 x}{5 x - 6})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{6 (- \frac{8 x}{5 x - 6})}{5 (- \frac{8 x}{5 x - 6}) + 8}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Multiply $- 1$ by $6$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{- 6 \frac{8 x}{5 x - 6}}{5 (- \frac{8 x}{5 x - 6}) + 8}$

Step 4.2.3.2

Combine $- 6$ and $\frac{8 x}{5 x - 6}$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{5 (- \frac{8 x}{5 x - 6}) + 8}$

Step 4.2.4

Simplify the denominator.

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Step 4.2.4.1

Multiply $5 (- \frac{8 x}{5 x - 6})$ .

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Step 4.2.4.1.1

Multiply $- 1$ by $5$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{- 5 \frac{8 x}{5 x - 6} + 8}$

Step 4.2.4.1.2

Combine $- 5$ and $\frac{8 x}{5 x - 6}$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{\frac{- 5 (8 x)}{5 x - 6} + 8}$

Step 4.2.4.1.3

Multiply $8$ by $- 5$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{\frac{- 40 x}{5 x - 6} + 8}$

Step 4.2.4.2

Move the negative in front of the fraction.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{- \frac{40 x}{5 x - 6} + 8}$

Step 4.2.4.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{5 x - 6}{5 x - 6}$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{- \frac{40 x}{5 x - 6} + \frac{8 (5 x - 6)}{5 x - 6}}$

Step 4.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{\frac{- 40 x + 8 (5 x - 6)}{5 x - 6}}$

Step 4.2.4.5

Rewrite $\frac{- 40 x + 8 (5 x - 6)}{5 x - 6}$ in a factored form.

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Step 4.2.4.5.1

Factor $8$ out of $- 40 x + 8 (5 x - 6)$ .

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Step 4.2.4.5.1.1

Factor $8$ out of $- 40 x$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{\frac{8 (- 5 x) + 8 (5 x - 6)}{5 x - 6}}$

Step 4.2.4.5.1.2

Factor $8$ out of $8 (- 5 x) + 8 (5 x - 6)$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{\frac{8 (- 5 x + 5 x - 6)}{5 x - 6}}$

Step 4.2.4.5.2

Add $- 5 x$ and $5 x$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{\frac{8 (0 - 6)}{5 x - 6}}$

Step 4.2.4.5.3

Subtract $6$ from $0$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{\frac{8 \cdot - 6}{5 x - 6}}$

Step 4.2.4.6

Multiply $8$ by $- 6$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{\frac{- 48}{5 x - 6}}$

Step 4.2.4.7

Move the negative in front of the fraction.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{- \frac{48}{5 x - 6}}$

Step 4.2.5

Multiply $- 6$ by $8$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 48 x}{5 x - 6}}{- \frac{48}{5 x - 6}}$

Step 4.2.6

Reduce the expression by cancelling the common factors.

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Step 4.2.6.1

Move the negative in front of the fraction.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{- \frac{48 x}{5 x - 6}}{- \frac{48}{5 x - 6}}$

Step 4.2.6.2

Dividing two negative values results in a positive value.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{48 x}{5 x - 6}}{\frac{48}{5 x - 6}}$

Step 4.2.7

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{48 x}{5 x - 6} \cdot \frac{5 x - 6}{48}$

Step 4.2.8

Cancel the common factor of $48$ .

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Step 4.2.8.1

Factor $48$ out of $48 x$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{48 (x)}{5 x - 6} \cdot \frac{5 x - 6}{48}$

Step 4.2.8.2

Cancel the common factor.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{48 x}{5 x - 6} \cdot \frac{5 x - 6}{48}$

Step 4.2.8.3

Rewrite the expression.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{x}{5 x - 6} \cdot (5 x - 6)$

Step 4.2.9

Cancel the common factor of $5 x - 6$ .

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Step 4.2.9.1

Cancel the common factor.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{x}{5 x - 6} \cdot (5 x - 6)$

Step 4.2.9.2

Rewrite the expression.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = x$

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 $f (\frac{6 x}{5 x + 8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{8 (\frac{6 x}{5 x + 8})}{5 (\frac{6 x}{5 x + 8}) - 6}$

Step 4.3.3

Combine $8$ and $\frac{6 x}{5 x + 8}$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{5 (\frac{6 x}{5 x + 8}) - 6}$

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Multiply $5 \frac{6 x}{5 x + 8}$ .

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Step 4.3.4.1.1

Combine $5$ and $\frac{6 x}{5 x + 8}$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{5 (6 x)}{5 x + 8} - 6}$

Step 4.3.4.1.2

Multiply $6$ by $5$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{30 x}{5 x + 8} - 6}$

Step 4.3.4.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{5 x + 8}{5 x + 8}$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{30 x}{5 x + 8} - 6 \cdot \frac{5 x + 8}{5 x + 8}}$

Step 4.3.4.3

Combine $- 6$ and $\frac{5 x + 8}{5 x + 8}$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{30 x}{5 x + 8} + \frac{- 6 (5 x + 8)}{5 x + 8}}$

Step 4.3.4.4

Combine the numerators over the common denominator.

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{30 x - 6 (5 x + 8)}{5 x + 8}}$

Step 4.3.4.5

Rewrite $\frac{30 x - 6 (5 x + 8)}{5 x + 8}$ in a factored form.

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Step 4.3.4.5.1

Factor $6$ out of $30 x - 6 (5 x + 8)$ .

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Step 4.3.4.5.1.1

Factor $6$ out of $30 x$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{6 (5 x) - 6 (5 x + 8)}{5 x + 8}}$

Step 4.3.4.5.1.2

Factor $6$ out of $- 6 (5 x + 8)$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{6 (5 x) + 6 (- (5 x + 8))}{5 x + 8}}$

Step 4.3.4.5.1.3

Factor $6$ out of $6 (5 x) + 6 (- (5 x + 8))$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{6 (5 x - (5 x + 8))}{5 x + 8}}$

Step 4.3.4.5.2

Apply the distributive property.

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{6 (5 x - (5 x) - 1 \cdot 8)}{5 x + 8}}$

Step 4.3.4.5.3

Multiply $5$ by $- 1$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{6 (5 x - 5 x - 1 \cdot 8)}{5 x + 8}}$

Step 4.3.4.5.4

Multiply $- 1$ by $8$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{6 (5 x - 5 x - 8)}{5 x + 8}}$

Step 4.3.4.5.5

Subtract $5 x$ from $5 x$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{6 (0 - 8)}{5 x + 8}}$

Step 4.3.4.5.6

Subtract $8$ from $0$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{6 \cdot - 8}{5 x + 8}}$

Step 4.3.4.6

Multiply $6$ by $- 8$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{- 48}{5 x + 8}}$

Step 4.3.4.7

Move the negative in front of the fraction.

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{- \frac{48}{5 x + 8}}$

Step 4.3.5

Multiply $8$ by $6$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{48 x}{5 x + 8}}{- \frac{48}{5 x + 8}}$

Step 4.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{6 x}{5 x + 8}) = - (\frac{48 x}{5 x + 8} \cdot (- \frac{5 x + 8}{48}))$

Step 4.3.7

Rewrite using the commutative property of multiplication.

$f (\frac{6 x}{5 x + 8}) = - (- \frac{48 x}{5 x + 8} \cdot \frac{5 x + 8}{48})$

Step 4.3.8

Cancel the common factor of $48$ .

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Step 4.3.8.1

Move the leading negative in $- \frac{48 x}{5 x + 8}$ into the numerator.

$f (\frac{6 x}{5 x + 8}) = - (\frac{- 48 x}{5 x + 8} \cdot \frac{5 x + 8}{48})$

Step 4.3.8.2

Factor $48$ out of $- 48 x$ .

$f (\frac{6 x}{5 x + 8}) = - (\frac{48 (- x)}{5 x + 8} \cdot \frac{5 x + 8}{48})$

Step 4.3.8.3

Cancel the common factor.

$f (\frac{6 x}{5 x + 8}) = - (\frac{48 (- x)}{5 x + 8} \cdot \frac{5 x + 8}{48})$

Step 4.3.8.4

Rewrite the expression.

$f (\frac{6 x}{5 x + 8}) = - (\frac{- x}{5 x + 8} \cdot (5 x + 8))$

Step 4.3.9

Cancel the common factor of $5 x + 8$ .

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Step 4.3.9.1

Cancel the common factor.

$f (\frac{6 x}{5 x + 8}) = - (\frac{- x}{5 x + 8} \cdot (5 x + 8))$

Step 4.3.9.2

Rewrite the expression.

$f (\frac{6 x}{5 x + 8}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6 x}{5 x + 8}$ is the inverse of $f (x) = - \frac{8 x}{5 x - 6}$ .

$f^{- 1} (x) = \frac{6 x}{5 x + 8}$