Algebra Examples Rational Expressions and Equations

Solve Over the Interval

$2 (x^{2} - 1) = 16$ , $(0, 4)$

Step 1

Divide each term in $2 (x^{2} - 1) = 16$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 (x^{2} - 1) = 16$ by $2$ .

$\frac{2 (x^{2} - 1)}{2} = \frac{16}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} - 1)}{2} = \frac{16}{2}$

Step 1.2.1.2

Divide $x^{2} - 1$ by $1$ .

$x^{2} - 1 = \frac{16}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $16$ by $2$ .

$x^{2} - 1 = 8$

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Add $1$ to both sides of the equation.

$x^{2} = 8 + 1$

Step 2.2

Add $8$ and $1$ .

$x^{2} = 9$

Step 3

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

$x = \pm \sqrt{9}$

Step 4

Simplify $\pm \sqrt{9}$ .

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Step 4.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 4.2

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

$x = \pm 3$

Step 5

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

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Step 5.1

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

$x = 3$

Step 5.2

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

$x = - 3$

Step 5.3

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

$x = 3, - 3$

Step 6

Find the values of $n$ that produce a value within the interval $(0, 4)$ .

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Step 6.1

The interval $(0, 4)$ does not contain $- 3$ . It is not part of the final solution.

$- 3$ is not on the interval

Step 6.2

The interval $(0, 4)$ contains $3$ .

$x = 3$

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