Precalculus Examples

$\frac{y^{2} - 5 y + 4}{y^{2} - 1} \div \frac{y^{2} - 9}{y^{2} + 5 y + 4}$

Step 1

Set the denominator in $\frac{y^{2} - 5 y + 4}{y^{2} - 1}$ equal to $0$ to find where the expression is undefined.

$y^{2} - 1 = 0$

Step 2

Solve for $y$ .

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Step 2.1

Add $1$ to both sides of the equation.

$y^{2} = 1$

Step 2.2

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

$y = \pm \sqrt{1}$

Step 2.3

Any root of $1$ is $1$ .

$y = \pm 1$

Step 2.4

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

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Step 2.4.1

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

$y = 1$

Step 2.4.2

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

$y = - 1$

Step 2.4.3

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 $\frac{y^{2} - 9}{y^{2} + 5 y + 4}$ equal to $0$ to find where the expression is undefined.

$y^{2} + 5 y + 4 = 0$

Step 4

Solve for $y$ .

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Step 4.1

Factor $y^{2} + 5 y + 4$ using the AC method.

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Step 4.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $5$ .

$1, 4$

Step 4.1.2

Write the factored form using these integers.

$(y + 1) (y + 4) = 0$

Step 4.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y + 1 = 0$

$y + 4 = 0$

Step 4.3

Set $y + 1$ equal to $0$ and solve for $y$ .

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Step 4.3.1

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Step 4.3.2

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 4.4

Set $y + 4$ equal to $0$ and solve for $y$ .

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Step 4.4.1

Set $y + 4$ equal to $0$ .

$y + 4 = 0$

Step 4.4.2

Subtract $4$ from both sides of the equation.

$y = - 4$

Step 4.5

The final solution is all the values that make $(y + 1) (y + 4) = 0$ true.

$y = - 1, - 4$

Step 5

Set the denominator in $\frac{y^{2} - 5 y + 4}{y^{2} - 1} \div \frac{y^{2} - 9}{y^{2} + 5 y + 4}$ equal to $0$ to find where the expression is undefined.

$\frac{y^{2} - 9}{y^{2} + 5 y + 4} = 0$

Step 6

Solve for $y$ .

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Step 6.1

Set the numerator equal to zero.

$y^{2} - 9 = 0$

Step 6.2

Solve the equation for $y$ .

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Step 6.2.1

Add $9$ to both sides of the equation.

$y^{2} = 9$

Step 6.2.2

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

$y = \pm \sqrt{9}$

Step 6.2.3

Simplify $\pm \sqrt{9}$ .

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Step 6.2.3.1

Rewrite $9$ as $3^{2}$ .

$y = \pm \sqrt{3^{2}}$

Step 6.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm 3$

Step 6.2.4

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

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Step 6.2.4.1

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

$y = 3$

Step 6.2.4.2

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

$y = - 3$

Step 6.2.4.3

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

$y = 3, - 3$

Step 7

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

Interval Notation:

$(- \infty, - 4) \cup (- 4, - 3) \cup (- 3, - 1) \cup (- 1, 1) \cup (1, 3) \cup (3, \infty)$

Set-Builder Notation:

${y | y \neq 1, - 1, 3, - 3, - 4}$

Step 8