Precalculus Examples

$\frac{x^{2} - 1}{5 x^{2} + 5 x}$

Step 1

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

$5 x^{2} + 5 x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $5 x$ out of $5 x^{2} + 5 x$ .

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Step 2.1.1

Factor $5 x$ out of $5 x^{2}$ .

$5 x (x) + 5 x = 0$

Step 2.1.2

Factor $5 x$ out of $5 x$ .

$5 x (x) + 5 x (1) = 0$

Step 2.1.3

Factor $5 x$ out of $5 x (x) + 5 x (1)$ .

$5 x (x + 1) = 0$

Step 2.2

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

$x = 0$

$x + 1 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

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

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Step 2.4.1

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

$x + 1 = 0$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

The final solution is all the values that make $5 x (x + 1) = 0$ true.

$x = 0, - 1$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

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

Step 4