Find the Domain y=( square root of 2x+14)/(x^2-49)

$y = \frac{\sqrt{2 x + 14}}{x^{2} - 49}$

Step 1

Set the radicand in $\sqrt{2 x + 14}$ greater than or equal to $0$ to find where the expression is defined.

$2 x + 14 \geq 0$

Step 2

Solve for $x$ .

Subtract $14$ from both sides of the inequality.

$2 x \geq - 14$

Divide each term in $2 x \geq - 14$ by $2$ and simplify.

Divide each term in $2 x \geq - 14$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 14}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 14}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 14}{2}$

Simplify the right side.

Divide $- 14$ by $2$ .

$x \geq - 7$

Step 3

Set the denominator in $\frac{\sqrt{2 x + 14}}{x^{2} - 49}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 49 = 0$

Step 4

Solve for $x$ .

Add $49$ to both sides of the equation.

$x^{2} = 49$

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

$x = \pm \sqrt{49}$

Simplify $\pm \sqrt{49}$ .

Rewrite $49$ as $7^{2}$ .

$x = \pm \sqrt{7^{2}}$

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

$x = \pm 7$

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

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

$x = 7$

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

$x = - 7$

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

$x = 7, - 7$

Step 5

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

Interval Notation:

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

Set-Builder Notation:

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

Step 6

Find the Domain y=( square root of 2x+14)/(x^2-49)

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