Algebra Examples

$2 x - \frac{3 x}{4}$

Step 1

Interchange the variables.

$x = 2 y - \frac{3 y}{4}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $2 y - \frac{3 y}{4} = x$ .

$2 y - \frac{3 y}{4} = x$

Step 2.2

Simplify $2 y - \frac{3 y}{4}$ .

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Step 2.2.1

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 y \cdot \frac{4}{4} - \frac{3 y}{4} = x$

Step 2.2.2

Simplify terms.

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Step 2.2.2.1

Combine $2 y$ and $\frac{4}{4}$ .

$\frac{2 y \cdot 4}{4} - \frac{3 y}{4} = x$

Step 2.2.2.2

Combine the numerators over the common denominator.

$\frac{2 y \cdot 4 - 3 y}{4} = x$

Step 2.2.3

Simplify the numerator.

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Step 2.2.3.1

Factor $y$ out of $2 y \cdot 4 - 3 y$ .

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Step 2.2.3.1.1

Factor $y$ out of $2 y \cdot 4$ .

$\frac{y (2 \cdot 4) - 3 y}{4} = x$

Step 2.2.3.1.2

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

$\frac{y (2 \cdot 4) + y \cdot - 3}{4} = x$

Step 2.2.3.1.3

Factor $y$ out of $y (2 \cdot 4) + y \cdot - 3$ .

$\frac{y (2 \cdot 4 - 3)}{4} = x$

Step 2.2.3.2

Multiply $2$ by $4$ .

$\frac{y (8 - 3)}{4} = x$

Step 2.2.3.3

Subtract $3$ from $8$ .

$\frac{y \cdot 5}{4} = x$

Step 2.2.4

Move $5$ to the left of $y$ .

$\frac{5 y}{4} = x$

Step 2.3

Multiply both sides of the equation by $\frac{4}{5}$ .

$\frac{4}{5} \cdot \frac{5 y}{4} = \frac{4}{5} x$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify $\frac{4}{5} \cdot \frac{5 y}{4}$ .

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Step 2.4.1.1.1

Cancel the common factor of $4$ .

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Step 2.4.1.1.1.1

Cancel the common factor.

$\frac{4}{5} \cdot \frac{5 y}{4} = \frac{4}{5} x$

Step 2.4.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 y) = \frac{4}{5} x$

Step 2.4.1.1.2

Cancel the common factor of $5$ .

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Step 2.4.1.1.2.1

Factor $5$ out of $5 y$ .

$\frac{1}{5} (5 (y)) = \frac{4}{5} x$

Step 2.4.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 y) = \frac{4}{5} x$

Step 2.4.1.1.2.3

Rewrite the expression.

$y = \frac{4}{5} x$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Combine $\frac{4}{5}$ and $x$ .

$y = \frac{4 x}{5}$

Step 3

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

$f^{- 1} (x) = \frac{4 x}{5}$

Step 4

Verify if $f^{- 1} (x) = \frac{4 x}{5}$ is the inverse of $f (x) = 2 x - \frac{3 x}{4}$ .

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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 $f^{- 1} (2 x - \frac{3 x}{4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{4 (2 x - \frac{3 x}{4})}{5}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

To write $2 x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{4 (2 x \cdot \frac{4}{4} - \frac{3 x}{4})}{5}$

Step 4.2.3.2

Combine $2 x$ and $\frac{4}{4}$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{4 (\frac{2 x \cdot 4}{4} - \frac{3 x}{4})}{5}$

Step 4.2.3.3

Combine the numerators over the common denominator.

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{4 (\frac{2 x \cdot 4 - 3 x}{4})}{5}$

Step 4.2.3.4

Rewrite $\frac{2 x \cdot 4 - 3 x}{4}$ in a factored form.

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Step 4.2.3.4.1

Factor $x$ out of $2 x \cdot 4 - 3 x$ .

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Step 4.2.3.4.1.1

Factor $x$ out of $2 x \cdot 4$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{4 (\frac{x (2 \cdot 4) - 3 x}{4})}{5}$

Step 4.2.3.4.1.2

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

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{4 (\frac{x (2 \cdot 4) + x \cdot - 3}{4})}{5}$

Step 4.2.3.4.1.3

Factor $x$ out of $x (2 \cdot 4) + x \cdot - 3$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{4 (\frac{x (2 \cdot 4 - 3)}{4})}{5}$

Step 4.2.3.4.2

Multiply $2$ by $4$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{4 (\frac{x (8 - 3)}{4})}{5}$

Step 4.2.3.4.3

Subtract $3$ from $8$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{4 (\frac{x \cdot 5}{4})}{5}$

Step 4.2.3.5

Move $5$ to the left of $x$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{4 (\frac{5 x}{4})}{5}$

Step 4.2.4

Combine $4$ and $\frac{5 x}{4}$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{\frac{4 (5 x)}{4}}{5}$

Step 4.2.5

Multiply $4$ by $5$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{\frac{20 x}{4}}{5}$

Step 4.2.6

Reduce the expression by cancelling the common factors.

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Step 4.2.6.1

Reduce the expression $\frac{20 x}{4}$ by cancelling the common factors.

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Step 4.2.6.1.1

Factor $4$ out of $20 x$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{\frac{4 (5 x)}{4}}{5}$

Step 4.2.6.1.2

Factor $4$ out of $4$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{\frac{4 (5 x)}{4 (1)}}{5}$

Step 4.2.6.1.3

Cancel the common factor.

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{\frac{4 (5 x)}{4 \cdot 1}}{5}$

Step 4.2.6.1.4

Rewrite the expression.

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{\frac{5 x}{1}}{5}$

Step 4.2.6.2

Divide $5 x$ by $1$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{5 x}{5}$

Step 4.2.7

Cancel the common factor of $5$ .

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Step 4.2.7.1

Cancel the common factor.

$f^{- 1} (2 x - \frac{3 x}{4}) = \frac{5 x}{5}$

Step 4.2.7.2

Divide $x$ by $1$ .

$f^{- 1} (2 x - \frac{3 x}{4}) = x$

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 $f (\frac{4 x}{5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{4 x}{5}) = 2 (\frac{4 x}{5}) - \frac{3 (\frac{4 x}{5})}{4}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Multiply $2 (\frac{4 x}{5})$ .

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Step 4.3.3.1.1

Combine $2$ and $\frac{4 x}{5}$ .

$f (\frac{4 x}{5}) = \frac{2 (4 x)}{5} - \frac{3 (\frac{4 x}{5})}{4}$

Step 4.3.3.1.2

Multiply $4$ by $2$ .

$f (\frac{4 x}{5}) = \frac{8 x}{5} - \frac{3 (\frac{4 x}{5})}{4}$

Step 4.3.3.2

Combine $3$ and $\frac{4 x}{5}$ .

$f (\frac{4 x}{5}) = \frac{8 x}{5} - \frac{\frac{3 (4 x)}{5}}{4}$

Step 4.3.3.3

Multiply $3$ by $4$ .

$f (\frac{4 x}{5}) = \frac{8 x}{5} - \frac{\frac{12 x}{5}}{4}$

Step 4.3.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{4 x}{5}) = \frac{8 x}{5} - (\frac{12 x}{5} \cdot \frac{1}{4})$

Step 4.3.3.5

Cancel the common factor of $4$ .

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Step 4.3.3.5.1

Factor $4$ out of $12 x$ .

$f (\frac{4 x}{5}) = \frac{8 x}{5} - (\frac{4 (3 x)}{5} \cdot \frac{1}{4})$

Step 4.3.3.5.2

Cancel the common factor.

$f (\frac{4 x}{5}) = \frac{8 x}{5} - (\frac{4 (3 x)}{5} \cdot \frac{1}{4})$

Step 4.3.3.5.3

Rewrite the expression.

$f (\frac{4 x}{5}) = \frac{8 x}{5} - \frac{3 x}{5}$

Step 4.3.4

Simplify terms.

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Step 4.3.4.1

Combine the numerators over the common denominator.

$f (\frac{4 x}{5}) = \frac{8 x - 3 x}{5}$

Step 4.3.4.2

Subtract $3 x$ from $8 x$ .

$f (\frac{4 x}{5}) = \frac{5 x}{5}$

Step 4.3.4.3

Cancel the common factor of $5$ .

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Step 4.3.4.3.1

Cancel the common factor.

$f (\frac{4 x}{5}) = \frac{5 x}{5}$

Step 4.3.4.3.2

Divide $x$ by $1$ .

$f (\frac{4 x}{5}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{4 x}{5}$ is the inverse of $f (x) = 2 x - \frac{3 x}{4}$ .

$f^{- 1} (x) = \frac{4 x}{5}$