Precalculus Examples

$\frac{5 y^{2}}{1 - y^{2}} \div (1 - \frac{1}{1 - y})$

Step 1

Set the denominator in $\frac{5 y^{2}}{1 - y^{2}}$ equal to $0$ to find where the expression is undefined.

$1 - y^{2} = 0$

Step 2

Solve for $y$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

$- y^{2} = - 1$

Step 2.2

Divide each term in $- y^{2} = - 1$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- y^{2} = - 1$ by $- 1$ .

$\frac{- y^{2}}{- 1} = \frac{- 1}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{- 1}{- 1}$

Step 2.2.2.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 1}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 1$ by $- 1$ .

$y^{2} = 1$

Step 2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{1}$

Step 2.4

Any root of $1$ is $1$ .

$y = \pm 1$

Step 2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = 1$

Step 2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - 1$

Step 2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = 1, - 1$

Step 3

Set the denominator in $\frac{1}{1 - y}$ equal to $0$ to find where the expression is undefined.

$1 - y = 0$

Step 4

Solve for $y$ .

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Step 4.1

Subtract $1$ from both sides of the equation.

$- y = - 1$

Step 4.2

Divide each term in $- y = - 1$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- y = - 1$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 1}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 1}{- 1}$

Step 4.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 1}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 1$ by $- 1$ .

$y = 1$

Step 5

Set the denominator in $\frac{5 y^{2}}{1 - y^{2}} \div (1 - \frac{1}{1 - y})$ equal to $0$ to find where the expression is undefined.

$1 - \frac{1}{1 - y} = 0$

Step 6

Solve for $y$ .

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Step 6.1

Subtract $1$ from both sides of the equation.

$- \frac{1}{1 - y} = - 1$

Step 6.2

Find the LCD of the terms in the equation.

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Step 6.2.1

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

$1 - y, 1$

Step 6.2.2

Remove parentheses.

$1 - y, 1$

Step 6.2.3

The LCM of one and any expression is the expression.

$1 - y$

Step 6.3

Multiply each term in $- \frac{1}{1 - y} = - 1$ by $1 - y$ to eliminate the fractions.

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Step 6.3.1

Multiply each term in $- \frac{1}{1 - y} = - 1$ by $1 - y$ .

$- \frac{1}{1 - y} (1 - y) = - (1 - y)$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $1 - y$ .

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Step 6.3.2.1.1

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

$\frac{- 1}{1 - y} (1 - y) = - (1 - y)$

Step 6.3.2.1.2

Cancel the common factor.

$\frac{- 1}{1 - y} (1 - y) = - (1 - y)$

Step 6.3.2.1.3

Rewrite the expression.

$- 1 = - (1 - y)$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Apply the distributive property.

$- 1 = - 1 \cdot 1 - - y$

Step 6.3.3.2

Multiply $- 1$ by $1$ .

$- 1 = - 1 - - y$

Step 6.3.3.3

Multiply $- - y$ .

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Step 6.3.3.3.1

Multiply $- 1$ by $- 1$ .

$- 1 = - 1 + 1 y$

Step 6.3.3.3.2

Multiply $y$ by $1$ .

$- 1 = - 1 + y$

Step 6.4

Solve the equation.

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Step 6.4.1

Rewrite the equation as $- 1 + y = - 1$ .

$- 1 + y = - 1$

Step 6.4.2

Move all terms not containing $y$ to the right side of the equation.

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Step 6.4.2.1

Add $1$ to both sides of the equation.

$y = - 1 + 1$

Step 6.4.2.2

Add $- 1$ and $1$ .

$y = 0$

Step 7

The domain is all values of $y$ that make the expression defined.

Interval Notation:

$(- \infty, - 1) \cup (- 1, 0) \cup (0, 1) \cup (1, \infty)$

Set-Builder Notation:

${y | y \neq 0, 1, - 1}$

Step 8