Algebra Examples

$\frac{6 x - 7}{2 x - 2}$

Step 1

Interchange the variables.

$x = \frac{6 y - 7}{2 y - 2}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{6 y - 7}{2 y - 2} = x$ .

$\frac{6 y - 7}{2 y - 2} = x$

Step 2.2

Factor $2$ out of $2 y - 2$ .

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Step 2.2.1

Factor $2$ out of $2 y$ .

$\frac{6 y - 7}{2 (y) - 2} = x$

Step 2.2.2

Factor $2$ out of $- 2$ .

$\frac{6 y - 7}{2 (y) + 2 (- 1)} = x$

Step 2.2.3

Factor $2$ out of $2 (y) + 2 (- 1)$ .

$\frac{6 y - 7}{2 (y - 1)} = x$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 (y - 1), 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$2 (y - 1)$

Step 2.4

Multiply each term in $\frac{6 y - 7}{2 (y - 1)} = x$ by $2 (y - 1)$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{6 y - 7}{2 (y - 1)} = x$ by $2 (y - 1)$ .

$\frac{6 y - 7}{2 (y - 1)} (2 (y - 1)) = x (2 (y - 1))$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Rewrite using the commutative property of multiplication.

$2 \frac{6 y - 7}{2 (y - 1)} (y - 1) = x (2 (y - 1))$

Step 2.4.2.2

Cancel the common factor of $2$ .

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Step 2.4.2.2.1

Cancel the common factor.

$2 \frac{6 y - 7}{2 (y - 1)} (y - 1) = x (2 (y - 1))$

Step 2.4.2.2.2

Rewrite the expression.

$\frac{6 y - 7}{y - 1} (y - 1) = x (2 (y - 1))$

Step 2.4.2.3

Cancel the common factor of $y - 1$ .

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Step 2.4.2.3.1

Cancel the common factor.

$\frac{6 y - 7}{y - 1} (y - 1) = x (2 (y - 1))$

Step 2.4.2.3.2

Rewrite the expression.

$6 y - 7 = x (2 (y - 1))$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Rewrite using the commutative property of multiplication.

$6 y - 7 = 2 x (y - 1)$

Step 2.4.3.2

Apply the distributive property.

$6 y - 7 = 2 x y + 2 x \cdot - 1$

Step 2.4.3.3

Multiply $- 1$ by $2$ .

$6 y - 7 = 2 x y - 2 x$

Step 2.5

Solve the equation.

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Step 2.5.1

Subtract $2 x y$ from both sides of the equation.

$6 y - 7 - 2 x y = - 2 x$

Step 2.5.2

Add $7$ to both sides of the equation.

$6 y - 2 x y = - 2 x + 7$

Step 2.5.3

Factor $2 y$ out of $6 y - 2 x y$ .

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Step 2.5.3.1

Factor $2 y$ out of $6 y$ .

$2 y (3) - 2 x y = - 2 x + 7$

Step 2.5.3.2

Factor $2 y$ out of $- 2 x y$ .

$2 y (3) + 2 y (- x) = - 2 x + 7$

Step 2.5.3.3

Factor $2 y$ out of $2 y (3) + 2 y (- x)$ .

$2 y (3 - x) = - 2 x + 7$

Step 2.5.4

Divide each term in $2 y (3 - x) = - 2 x + 7$ by $2 (3 - x)$ and simplify.

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Step 2.5.4.1

Divide each term in $2 y (3 - x) = - 2 x + 7$ by $2 (3 - x)$ .

$\frac{2 y (3 - x)}{2 (3 - x)} = \frac{- 2 x}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $2$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

$\frac{2 y (3 - x)}{2 (3 - x)} = \frac{- 2 x}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Step 2.5.4.2.1.2

Rewrite the expression.

$\frac{y (3 - x)}{3 - x} = \frac{- 2 x}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Step 2.5.4.2.2

Cancel the common factor of $3 - x$ .

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Step 2.5.4.2.2.1

Cancel the common factor.

$\frac{y (3 - x)}{3 - x} = \frac{- 2 x}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Step 2.5.4.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 2 x}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Simplify each term.

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Step 2.5.4.3.1.1

Cancel the common factor of $- 2$ and $2$ .

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Step 2.5.4.3.1.1.1

Factor $2$ out of $- 2 x$ .

$y = \frac{2 (- x)}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Step 2.5.4.3.1.1.2

Cancel the common factors.

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Step 2.5.4.3.1.1.2.1

Cancel the common factor.

$y = \frac{2 (- x)}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Step 2.5.4.3.1.1.2.2

Rewrite the expression.

$y = \frac{- x}{3 - x} + \frac{7}{2 (3 - x)}$

Step 2.5.4.3.1.2

Move the negative in front of the fraction.

$y = - \frac{x}{3 - x} + \frac{7}{2 (3 - x)}$

Step 2.5.4.3.2

To write $- \frac{x}{3 - x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{x}{3 - x} \cdot \frac{2}{2} + \frac{7}{2 (3 - x)}$

Step 2.5.4.3.3

Write each expression with a common denominator of $(3 - x) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.4.3.3.1

Multiply $\frac{x}{3 - x}$ by $\frac{2}{2}$ .

$y = - \frac{x \cdot 2}{(3 - x) \cdot 2} + \frac{7}{2 (3 - x)}$

Step 2.5.4.3.3.2

Reorder the factors of $(3 - x) \cdot 2$ .

$y = - \frac{x \cdot 2}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Step 2.5.4.3.4

Combine the numerators over the common denominator.

$y = \frac{- x \cdot 2 + 7}{2 (3 - x)}$

Step 2.5.4.3.5

Multiply $2$ by $- 1$ .

$y = \frac{- 2 x + 7}{2 (3 - x)}$

Step 2.5.4.3.6

Factor $- 1$ out of $- 2 x$ .

$y = \frac{- (2 x) + 7}{2 (3 - x)}$

Step 2.5.4.3.7

Rewrite $7$ as $- 1 (- 7)$ .

$y = \frac{- (2 x) - 1 (- 7)}{2 (3 - x)}$

Step 2.5.4.3.8

Factor $- 1$ out of $- (2 x) - 1 (- 7)$ .

$y = \frac{- (2 x - 7)}{2 (3 - x)}$

Step 2.5.4.3.9

Simplify the expression.

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Step 2.5.4.3.9.1

Rewrite $- (2 x - 7)$ as $- 1 (2 x - 7)$ .

$y = \frac{- 1 (2 x - 7)}{2 (3 - x)}$

Step 2.5.4.3.9.2

Move the negative in front of the fraction.

$y = - \frac{2 x - 7}{2 (3 - x)}$

Step 3

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

$f^{- 1} (x) = - \frac{2 x - 7}{2 (3 - x)}$

Step 4

Verify if $f^{- 1} (x) = - \frac{2 x - 7}{2 (3 - x)}$ is the inverse of $f (x) = \frac{6 x - 7}{2 x - 2}$ .

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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 $f^{- 1} (\frac{6 x - 7}{2 x - 2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 x - 2}) - 7}{2 (3 - (\frac{6 x - 7}{2 x - 2}))}$

Step 4.2.3

Simplify with factoring out.

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Step 4.2.3.1

Factor $2$ out of $2 x - 2$ .

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Step 4.2.3.1.1

Factor $2$ out of $2 x$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x) - 2}) - 7}{2 (3 - \frac{6 x - 7}{2 x - 2})}$

Step 4.2.3.1.2

Factor $2$ out of $- 2$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x) + 2 (- 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 x - 2})}$

Step 4.2.3.1.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x - 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 x - 2})}$

Step 4.2.3.2

Factor $2$ out of $2 x - 2$ .

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Step 4.2.3.2.1

Factor $2$ out of $2 x$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x - 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 (x) - 2})}$

Step 4.2.3.2.2

Factor $2$ out of $- 2$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x - 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 (x) + 2 (- 1)})}$

Step 4.2.3.2.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x - 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4

Simplify the numerator.

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Step 4.2.4.1

Cancel the common factor of $2$ .

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Step 4.2.4.1.1

Cancel the common factor.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x - 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4.1.2

Rewrite the expression.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7}{x - 1} - 7}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7}{x - 1} - 7 \cdot \frac{x - 1}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4.3

Combine $- 7$ and $\frac{x - 1}{x - 1}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7}{x - 1} + \frac{- 7 (x - 1)}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7 - 7 (x - 1)}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4.5

Rewrite $\frac{6 x - 7 - 7 (x - 1)}{x - 1}$ in a factored form.

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Step 4.2.4.5.1

Apply the distributive property.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7 - 7 x - 7 \cdot - 1}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4.5.2

Multiply $- 7$ by $- 1$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7 - 7 x + 7}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4.5.3

Subtract $7 x$ from $6 x$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{- x - 7 + 7}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4.5.4

Add $- 7$ and $7$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{- x + 0}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4.5.5

Add $- x$ and $0$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{- x}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.4.6

Move the negative in front of the fraction.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.5

Simplify the denominator.

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Step 4.2.5.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{2 (x - 1)}{2 (x - 1)}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (3 \cdot \frac{2 (x - 1)}{2 (x - 1)} - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.5.2

Combine $3$ and $\frac{2 (x - 1)}{2 (x - 1)}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{3 (2 (x - 1))}{2 (x - 1)} - \frac{6 x - 7}{2 (x - 1)})}$

Step 4.2.5.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{3 (2 (x - 1)) - (6 x - 7)}{2 (x - 1)})}$

Step 4.2.5.4

Rewrite $\frac{3 (2 (x - 1)) - (6 x - 7)}{2 (x - 1)}$ in a factored form.

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Step 4.2.5.4.1

Apply the distributive property.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{3 (2 x + 2 \cdot - 1) - (6 x - 7)}{2 (x - 1)})}$

Step 4.2.5.4.2

Multiply $2$ by $- 1$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{3 (2 x - 2) - (6 x - 7)}{2 (x - 1)})}$

Step 4.2.5.4.3

Apply the distributive property.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{3 (2 x) + 3 \cdot - 2 - (6 x - 7)}{2 (x - 1)})}$

Step 4.2.5.4.4

Multiply $2$ by $3$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{6 x + 3 \cdot - 2 - (6 x - 7)}{2 (x - 1)})}$

Step 4.2.5.4.5

Multiply $3$ by $- 2$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{6 x - 6 - (6 x - 7)}{2 (x - 1)})}$

Step 4.2.5.4.6

Apply the distributive property.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{6 x - 6 - (6 x) + 7}{2 (x - 1)})}$

Step 4.2.5.4.7

Multiply $6$ by $- 1$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{6 x - 6 - 6 x + 7}{2 (x - 1)})}$

Step 4.2.5.4.8

Multiply $- 1$ by $- 7$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{6 x - 6 - 6 x + 7}{2 (x - 1)})}$

Step 4.2.5.4.9

Subtract $6 x$ from $6 x$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{0 - 6 + 7}{2 (x - 1)})}$

Step 4.2.5.4.10

Subtract $6$ from $0$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{- 6 + 7}{2 (x - 1)})}$

Step 4.2.5.4.11

Add $- 6$ and $7$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{1}{2 (x - 1)})}$

Step 4.2.6

Simplify terms.

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Step 4.2.6.1

Combine $2$ and $\frac{1}{2 (x - 1)}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{\frac{2}{2 (x - 1)}}$

Step 4.2.6.2

Reduce the expression $\frac{2}{2 (x - 1)}$ by cancelling the common factors.

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Step 4.2.6.2.1

Cancel the common factor.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{\frac{2}{2 (x - 1)}}$

Step 4.2.6.2.2

Rewrite the expression.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{\frac{1}{x - 1}}$

Step 4.2.7

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - (- \frac{x}{x - 1} \cdot (x - 1))$

Step 4.2.8

Cancel the common factor of $x - 1$ .

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Step 4.2.8.1

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

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - (\frac{- x}{x - 1} \cdot (x - 1))$

Step 4.2.8.2

Cancel the common factor.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - (\frac{- x}{x - 1} \cdot (x - 1))$

Step 4.2.8.3

Rewrite the expression.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = x$

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 $f (- \frac{2 x - 7}{2 (3 - x)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{6 (- \frac{2 x - 7}{2 (3 - x)}) - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

Cancel the common factor of $2$ .

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Step 4.3.3.1.1

Move the leading negative in $- \frac{2 x - 7}{2 (3 - x)}$ into the numerator.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{6 (\frac{- (2 x - 7)}{2 (3 - x)}) - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.1.2

Factor $2$ out of $6$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{2 (3) (\frac{- (2 x - 7)}{2 (3 - x)}) - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.1.3

Cancel the common factor.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{2 \cdot (3 (\frac{- (2 x - 7)}{2 (3 - x)})) - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.1.4

Rewrite the expression.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{3 (\frac{- (2 x - 7)}{3 - x}) - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.2

Combine $3$ and $\frac{- (2 x - 7)}{3 - x}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{3 (- (2 x - 7))}{3 - x} - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.3

Multiply $- 1$ by $3$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- 3 (2 x - 7)}{3 - x} - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.4

Move the negative in front of the fraction.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{- \frac{3 (2 x - 7)}{3 - x} - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.5

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3 - x}{3 - x}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{- \frac{3 (2 x - 7)}{3 - x} - 7 \cdot \frac{3 - x}{3 - x}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.6

Combine $- 7$ and $\frac{3 - x}{3 - x}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{- \frac{3 (2 x - 7)}{3 - x} + \frac{- 7 (3 - x)}{3 - x}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.7

Combine the numerators over the common denominator.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- 3 (2 x - 7) - 7 (3 - x)}{3 - x}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.8

Reorder terms.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- 3 (2 x - 7) - 7 (3 - x)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9

Rewrite $\frac{- 3 (2 x - 7) - 7 (3 - x)}{- x + 3}$ in a factored form.

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Step 4.3.3.9.1

Factor $- 1$ out of $- 3 (2 x - 7) - 7 (3 - x)$ .

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Step 4.3.3.9.1.1

Reorder $3$ and $- x$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- 3 (2 x - 7) - 7 (- x + 3)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.1.2

Factor $- 1$ out of $- 3 (2 x - 7)$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (3 (2 x - 7)) - 7 (- x + 3)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.1.3

Factor $- 1$ out of $- 7 (- x + 3)$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (3 (2 x - 7)) - (7 (- x + 3))}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.1.4

Factor $- 1$ out of $- (3 (2 x - 7)) - (7 (- x + 3))$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (3 (2 x - 7) + 7 (- x + 3))}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.2

Apply the distributive property.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (3 (2 x) + 3 \cdot - 7 + 7 (- x + 3))}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.3

Multiply $2$ by $3$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (6 x + 3 \cdot - 7 + 7 (- x + 3))}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.4

Multiply $3$ by $- 7$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (6 x - 21 + 7 (- x + 3))}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.5

Apply the distributive property.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (6 x - 21 + 7 (- x) + 7 \cdot 3)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.6

Multiply $- 1$ by $7$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (6 x - 21 - 7 x + 7 \cdot 3)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.7

Multiply $7$ by $3$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (6 x - 21 - 7 x + 21)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.8

Subtract $7 x$ from $6 x$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (- x - 21 + 21)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.9

Add $- 21$ and $21$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (- x + 0)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.10

Add $- x$ and $0$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{1 x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.11

Combine exponents.

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Step 4.3.3.9.11.1

Factor out negative.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.11.2

Multiply $- 1$ by $- 1$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{1 x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.3.9.11.3

Multiply $x$ by $1$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Factor $2$ out of $2 (- \frac{2 x - 7}{2 (3 - x)}) - 2$ .

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Step 4.3.4.1.1

Factor $2$ out of $- 2$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) + 2 (- 1)}$

Step 4.3.4.1.2

Factor $2$ out of $2 (- \frac{2 x - 7}{2 (3 - x)}) + 2 (- 1)$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)} - 1)}$

Step 4.3.4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2 (3 - x)}{2 (3 - x)}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)} - 1 \cdot \frac{2 (3 - x)}{2 (3 - x)})}$

Step 4.3.4.3

Combine $- 1$ and $\frac{2 (3 - x)}{2 (3 - x)}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)} + \frac{- (2 (3 - x))}{2 (3 - x)})}$

Step 4.3.4.4

Combine the numerators over the common denominator.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7) - (2 (3 - x))}{2 (3 - x)})}$

Step 4.3.4.5

Rewrite $\frac{- (2 x - 7) - (2 (3 - x))}{2 (3 - x)}$ in a factored form.

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Step 4.3.4.5.1

Factor $- 1$ out of $- (2 x - 7) - 1 \cdot 2 (3 - x)$ .

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Step 4.3.4.5.1.1

Reorder $3$ and $- x$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7) - 1 \cdot (2 (- x + 3))}{2 (3 - x)})}$

Step 4.3.4.5.1.2

Factor $- 1$ out of $- 1 \cdot 2 (- x + 3)$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7) - (2 (- x + 3))}{2 (3 - x)})}$

Step 4.3.4.5.1.3

Factor $- 1$ out of $- (2 x - 7) - (2 (- x + 3))$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7 + 2 (- x + 3))}{2 (3 - x)})}$

Step 4.3.4.5.2

Apply the distributive property.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7 + 2 (- x) + 2 \cdot 3)}{2 (3 - x)})}$

Step 4.3.4.5.3

Multiply $- 1$ by $2$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7 - 2 x + 2 \cdot 3)}{2 (3 - x)})}$

Step 4.3.4.5.4

Multiply $2$ by $3$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7 - 2 x + 6)}{2 (3 - x)})}$

Step 4.3.4.5.5

Subtract $2 x$ from $2 x$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (0 - 7 + 6)}{2 (3 - x)})}$

Step 4.3.4.5.6

Subtract $7$ from $0$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (- 7 + 6)}{2 (3 - x)})}$

Step 4.3.4.5.7

Add $- 7$ and $6$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{1}{2 (3 - x)})}$

Step 4.3.4.6

Multiply $- 1$ by $- 1$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{1}{2 (3 - x)})}$

Step 4.3.5

Simplify terms.

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Step 4.3.5.1

Combine $2$ and $\frac{1}{2 (3 - x)}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{\frac{2}{2 (3 - x)}}$

Step 4.3.5.2

Reduce the expression $\frac{2}{2 (3 - x)}$ by cancelling the common factors.

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Step 4.3.5.2.1

Cancel the common factor.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{\frac{2}{2 (3 - x)}}$

Step 4.3.5.2.2

Rewrite the expression.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{\frac{1}{3 - x}}$

Step 4.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{x}{- x + 3} \cdot (3 - x)$

Step 4.3.7

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

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{x (3 - x)}{- x + 3}$

Step 4.3.8

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

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Step 4.3.8.1

Reorder terms.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{x (- x + 3)}{- x + 3}$

Step 4.3.8.2

Cancel the common factor.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{x (- x + 3)}{- x + 3}$

Step 4.3.8.3

Divide $x$ by $1$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{2 x - 7}{2 (3 - x)}$ is the inverse of $f (x) = \frac{6 x - 7}{2 x - 2}$ .

$f^{- 1} (x) = - \frac{2 x - 7}{2 (3 - x)}$