Precalculus Examples

$y = \frac{(x + 6) (x + 2) (x + 8)}{(x + 9) (x + 7)}$

Step 1

Set the denominator in $\frac{(x + 6) (x + 2) (x + 8)}{(x + 9) (x + 7)}$ equal to $0$ to find where the expression is undefined.

$(x + 9) (x + 7) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$x + 9 = 0$

$x + 7 = 0$

Step 2.2

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 2.2.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 2.3

Set $x + 7$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 2.3.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 2.4

The final solution is all the values that make $(x + 9) (x + 7) = 0$ true.

$x = - 9, - 7$

Step 3

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

Interval Notation:

$(- \infty, - 9) \cup (- 9, - 7) \cup (- 7, \infty)$

Set-Builder Notation:

${x | x \neq - 7, - 9}$

Step 4