Algebra Examples

$\frac{4}{11 - 2 x}$

Step 1

Interchange the variables.

$x = \frac{4}{11 - 2 y}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{4}{11 - 2 y} = x$ .

$\frac{4}{11 - 2 y} = x$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$11 - 2 y, 1$

Step 2.2.2

Remove parentheses.

$11 - 2 y, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$11 - 2 y$

Step 2.3

Multiply each term in $\frac{4}{11 - 2 y} = x$ by $11 - 2 y$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{4}{11 - 2 y} = x$ by $11 - 2 y$ .

$\frac{4}{11 - 2 y} (11 - 2 y) = x (11 - 2 y)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $11 - 2 y$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{4}{11 - 2 y} (11 - 2 y) = x (11 - 2 y)$

Step 2.3.2.1.2

Rewrite the expression.

$4 = x (11 - 2 y)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$4 = x \cdot 11 + x (- 2 y)$

Step 2.3.3.2

Reorder.

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Step 2.3.3.2.1

Move $11$ to the left of $x$ .

$4 = 11 \cdot x + x (- 2 y)$

Step 2.3.3.2.2

Rewrite using the commutative property of multiplication.

$4 = 11 x - 2 x y$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $11 x - 2 x y = 4$ .

$11 x - 2 x y = 4$

Step 2.4.2

Subtract $11 x$ from both sides of the equation.

$- 2 x y = 4 - 11 x$

Step 2.4.3

Divide each term in $- 2 x y = 4 - 11 x$ by $- 2 x$ and simplify.

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Step 2.4.3.1

Divide each term in $- 2 x y = 4 - 11 x$ by $- 2 x$ .

$\frac{- 2 x y}{- 2 x} = \frac{4}{- 2 x} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $- 2$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{- 2 x y}{- 2 x} = \frac{4}{- 2 x} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.2.1.2

Rewrite the expression.

$\frac{x y}{x} = \frac{4}{- 2 x} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.2.2

Cancel the common factor of $x$ .

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Step 2.4.3.2.2.1

Cancel the common factor.

$\frac{x y}{x} = \frac{4}{- 2 x} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.2.2.2

Divide $y$ by $1$ .

$y = \frac{4}{- 2 x} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Simplify each term.

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Step 2.4.3.3.1.1

Cancel the common factor of $4$ and $- 2$ .

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Step 2.4.3.3.1.1.1

Factor $2$ out of $4$ .

$y = \frac{2 (2)}{- 2 x} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.3.1.1.2

Cancel the common factors.

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Step 2.4.3.3.1.1.2.1

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

$y = \frac{2 (2)}{2 (- x)} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 2}{2 (- x)} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{2}{- x} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.3.1.2

Move the negative in front of the fraction.

$y = - \frac{2}{x} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.3.1.3

Cancel the common factor of $x$ .

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Step 2.4.3.3.1.3.1

Cancel the common factor.

$y = - \frac{2}{x} + \frac{- 11 x}{- 2 x}$

Step 2.4.3.3.1.3.2

Rewrite the expression.

$y = - \frac{2}{x} + \frac{- 11}{- 2}$

Step 2.4.3.3.1.4

Dividing two negative values results in a positive value.

$y = - \frac{2}{x} + \frac{11}{2}$

Step 3

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

$f^{- 1} (x) = - \frac{2}{x} + \frac{11}{2}$

Step 4

Verify if $f^{- 1} (x) = - \frac{2}{x} + \frac{11}{2}$ is the inverse of $f (x) = \frac{4}{11 - 2 x}$ .

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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 $f^{- 1} (\frac{4}{11 - 2 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = - \frac{2}{\frac{4}{11 - 2 x}} + \frac{11}{2}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{4}{11 - 2 x}) = - (2 (\frac{11 - 2 x}{4})) + \frac{11}{2}$

Step 4.2.3.2

Cancel the common factor of $2$ .

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Step 4.2.3.2.1

Factor $2$ out of $4$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = - (2 (\frac{11 - 2 x}{2 (2)})) + \frac{11}{2}$

Step 4.2.3.2.2

Cancel the common factor.

$f^{- 1} (\frac{4}{11 - 2 x}) = - (2 (\frac{11 - 2 x}{2 \cdot 2})) + \frac{11}{2}$

Step 4.2.3.2.3

Rewrite the expression.

$f^{- 1} (\frac{4}{11 - 2 x}) = - \frac{11 - 2 x}{2} + \frac{11}{2}$

Step 4.2.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{- (11 - 2 x) + 11}{2}$

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

Apply the distributive property.

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{- 1 \cdot 11 - (- 2 x) + 11}{2}$

Step 4.2.5.2

Multiply $- 1$ by $11$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{- 11 - (- 2 x) + 11}{2}$

Step 4.2.5.3

Multiply $- 2$ by $- 1$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{- 11 + 2 x + 11}{2}$

Step 4.2.6

Simplify terms.

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Step 4.2.6.1

Combine the opposite terms in $- 11 + 2 x + 11$ .

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Step 4.2.6.1.1

Add $- 11$ and $11$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{2 x + 0}{2}$

Step 4.2.6.1.2

Add $2 x$ and $0$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{2 x}{2}$

Step 4.2.6.2

Cancel the common factor of $2$ .

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Step 4.2.6.2.1

Cancel the common factor.

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{2 x}{2}$

Step 4.2.6.2.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = x$

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 $f (- \frac{2}{x} + \frac{11}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 - 2 (- \frac{2}{x} + \frac{11}{2})}$

Step 4.3.3

Simplify the denominator.

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Step 4.3.3.1

Apply the distributive property.

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 - 2 (- \frac{2}{x}) - 2 (\frac{11}{2})}$

Step 4.3.3.2

Multiply $- 2 (- \frac{2}{x})$ .

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Step 4.3.3.2.1

Multiply $- 1$ by $- 2$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + 2 (\frac{2}{x}) - 2 (\frac{11}{2})}$

Step 4.3.3.2.2

Combine $2$ and $\frac{2}{x}$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{2 \cdot 2}{x} - 2 (\frac{11}{2})}$

Step 4.3.3.2.3

Multiply $2$ by $2$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{4}{x} - 2 (\frac{11}{2})}$

Step 4.3.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.3.1

Factor $2$ out of $- 2$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{4}{x} + 2 (- 1) (\frac{11}{2})}$

Step 4.3.3.3.2

Cancel the common factor.

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{4}{x} + 2 \cdot (- 1 (\frac{11}{2}))}$

Step 4.3.3.3.3

Rewrite the expression.

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{4}{x} - 1 \cdot 11}$

Step 4.3.3.4

Multiply $- 1$ by $11$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{4}{x} - 11}$

Step 4.3.3.5

Subtract $11$ from $11$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{\frac{4}{x} + 0}$

Step 4.3.3.6

Add $\frac{4}{x}$ and $0$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{\frac{4}{x}}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{2}{x} + \frac{11}{2}) = 4 (\frac{x}{4})$

Step 4.3.5

Cancel the common factor of $4$ .

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Step 4.3.5.1

Cancel the common factor.

$f (- \frac{2}{x} + \frac{11}{2}) = 4 (\frac{x}{4})$

Step 4.3.5.2

Rewrite the expression.

$f (- \frac{2}{x} + \frac{11}{2}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{2}{x} + \frac{11}{2}$ is the inverse of $f (x) = \frac{4}{11 - 2 x}$ .

$f^{- 1} (x) = - \frac{2}{x} + \frac{11}{2}$