Finite Math Examples

$\sqrt{2 n^{2} - 1}$

Step 1

Set the radicand in $\sqrt{2 n^{2} - 1}$ less than $0$ to find where the expression is undefined.

$2 n^{2} - 1 < 0$

Step 2

Solve for $n$ .

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Step 2.1

Add $1$ to both sides of the inequality.

$2 n^{2} < 1$

Step 2.2

Divide each term in $2 n^{2} < 1$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 n^{2} < 1$ by $2$ .

$\frac{2 n^{2}}{2} < \frac{1}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 n^{2}}{2} < \frac{1}{2}$

Step 2.2.2.1.2

Divide $n^{2}$ by $1$ .

$n^{2} < \frac{1}{2}$

Step 2.3

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

$\sqrt{n^{2}} < \sqrt{\frac{1}{2}}$

Step 2.4

Simplify the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Pull terms out from under the radical.

$| n | < \sqrt{\frac{1}{2}}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{\frac{1}{2}}$ .

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Step 2.4.2.1.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$| n | < \frac{\sqrt{1}}{\sqrt{2}}$

Step 2.4.2.1.2

Any root of $1$ is $1$ .

$| n | < \frac{1}{\sqrt{2}}$

Step 2.4.2.1.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| n | < \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.4.2.1.4

Combine and simplify the denominator.

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Step 2.4.2.1.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| n | < \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.4.2.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$| n | < \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.4.2.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$| n | < \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.4.2.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| n | < \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.4.2.1.4.5

Add $1$ and $1$ .

$| n | < \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.4.2.1.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.4.2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$| n | < \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.4.2.1.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| n | < \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.4.2.1.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$| n | < \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.2.1.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.2.1.4.6.4.1

Cancel the common factor.

$| n | < \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.2.1.4.6.4.2

Rewrite the expression.

$| n | < \frac{\sqrt{2}}{2^{1}}$

Step 2.4.2.1.4.6.5

Evaluate the exponent.

$| n | < \frac{\sqrt{2}}{2}$

Step 2.5

Write $| n | < \frac{\sqrt{2}}{2}$ as a piecewise.

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Step 2.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$n \geq 0$

Step 2.5.2

In the piece where $n$ is non-negative, remove the absolute value.

$n < \frac{\sqrt{2}}{2}$

Step 2.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$n < 0$

Step 2.5.4

In the piece where $n$ is negative, remove the absolute value and multiply by $- 1$ .

$- n < \frac{\sqrt{2}}{2}$

Step 2.5.5

Write as a piecewise.

${\begin{cases} n < \frac{\sqrt{2}}{2} & n \geq 0 \\ - n < \frac{\sqrt{2}}{2} & n < 0 \end{cases}$

Step 2.6

Find the intersection of $n < \frac{\sqrt{2}}{2}$ and $n \geq 0$ .

$0 \leq n < \frac{\sqrt{2}}{2}$

Step 2.7

Solve $- n < \frac{\sqrt{2}}{2}$ when $n < 0$ .

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Step 2.7.1

Divide each term in $- n < \frac{\sqrt{2}}{2}$ by $- 1$ and simplify.

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Step 2.7.1.1

Divide each term in $- n < \frac{\sqrt{2}}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- n}{- 1} > \frac{\frac{\sqrt{2}}{2}}{- 1}$

Step 2.7.1.2

Simplify the left side.

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Step 2.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} > \frac{\frac{\sqrt{2}}{2}}{- 1}$

Step 2.7.1.2.2

Divide $n$ by $1$ .

$n > \frac{\frac{\sqrt{2}}{2}}{- 1}$

Step 2.7.1.3

Simplify the right side.

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Step 2.7.1.3.1

Move the negative one from the denominator of $\frac{\frac{\sqrt{2}}{2}}{- 1}$ .

$n > - 1 \cdot \frac{\sqrt{2}}{2}$

Step 2.7.1.3.2

Rewrite $- 1 \cdot \frac{\sqrt{2}}{2}$ as $- \frac{\sqrt{2}}{2}$ .

$n > - \frac{\sqrt{2}}{2}$

Step 2.7.2

Find the intersection of $n > - \frac{\sqrt{2}}{2}$ and $n < 0$ .

$- \frac{\sqrt{2}}{2} < n < 0$

Step 2.8

Find the union of the solutions.

$- \frac{\sqrt{2}}{2} < n < \frac{\sqrt{2}}{2}$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$- \frac{\sqrt{2}}{2} < n < \frac{\sqrt{2}}{2}$

$(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- \frac{\sqrt{2}}{2} < n < \frac{\sqrt{2}}{2} (- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Interval Notation:

$(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Step 5