Precalculus Examples

$y = \frac{(x - 7) (x + 10)}{(x + 10) (x - 3) (x - 7)}$

Step 1

Set the denominator in $\frac{(x - 7) (x + 10)}{(x + 10) (x - 3) (x - 7)}$ equal to $0$ to find where the expression is undefined.

$(x + 10) (x - 3) (x - 7) = 0$

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 $0$ .

$x + 10 = 0$

$x - 3 = 0$

$x - 7 = 0$

Step 2.2

Set $x + 10$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Step 2.2.2

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 2.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.4

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

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Step 2.4.1

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

$x - 7 = 0$

Step 2.4.2

Add $7$ to both sides of the equation.

$x = 7$

Step 2.5

The final solution is all the values that make $(x + 10) (x - 3) (x - 7) = 0$ true.

$x = - 10, 3, 7$

Step 3

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

Interval Notation:

$(- \infty, - 10) \cup (- 10, 3) \cup (3, 7) \cup (7, \infty)$

Set-Builder Notation:

${x | x \neq 3, 7, - 10}$

Step 4