Precalculus Examples

$\frac{4 x^{2}}{(x - 2) (x + 9)}$

Step 1

Set the denominator in $\frac{4 x^{2}}{(x - 2) (x + 9)}$ equal to $0$ to find where the expression is undefined.

$(x - 2) (x + 9) = 0$

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 $0$ .

$x - 2 = 0$

$x + 9 = 0$

Step 2.2

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.3

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

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Step 2.3.1

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

$x + 9 = 0$

Step 2.3.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 2.4

The final solution is all the values that make $(x - 2) (x + 9) = 0$ true.

$x = 2, - 9$

Step 3

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

Interval Notation:

$(- \infty, - 9) \cup (- 9, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2, - 9}$

Step 4