Finite Math Examples

$\sqrt{64 - 4 x^{2}}$

Step 1

Set the radicand in $\sqrt{64 - 4 x^{2}}$ less than $0$ to find where the expression is undefined.

$64 - 4 x^{2} < 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $64$ from both sides of the inequality.

$- 4 x^{2} < - 64$

Step 2.2

Divide each term in $- 4 x^{2} < - 64$ by $- 4$ and simplify.

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Step 2.2.1

Divide each term in $- 4 x^{2} < - 64$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x^{2}}{- 4} > \frac{- 64}{- 4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 4 x^{2}}{- 4} > \frac{- 64}{- 4}$

Step 2.2.2.1.2

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

$x^{2} > \frac{- 64}{- 4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 64$ by $- 4$ .

$x^{2} > 16$

Step 2.3

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

$\sqrt{x^{2}} > \sqrt{16}$

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.

$| x | > \sqrt{16}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{16}$ .

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Step 2.4.2.1.1

Rewrite $16$ as $4^{2}$ .

$| x | > \sqrt{4^{2}}$

Step 2.4.2.1.2

Pull terms out from under the radical.

$| x | > | 4 |$

Step 2.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$| x | > 4$

Step 2.5

Write $| x | > 4$ 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.

$x \geq 0$

Step 2.5.2

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

$x > 4$

Step 2.5.3

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

$x < 0$

Step 2.5.4

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

$- x > 4$

Step 2.5.5

Write as a piecewise.

${\begin{cases} x > 4 & x \geq 0 \\ - x > 4 & x < 0 \end{cases}$

Step 2.6

Find the intersection of $x > 4$ and $x \geq 0$ .

$x > 4$

Step 2.7

Divide each term in $- x > 4$ by $- 1$ and simplify.

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Step 2.7.1

Divide each term in $- x > 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{4}{- 1}$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{4}{- 1}$

Step 2.7.2.2

Divide $x$ by $1$ .

$x < \frac{4}{- 1}$

Step 2.7.3

Simplify the right side.

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Step 2.7.3.1

Divide $4$ by $- 1$ .

$x < - 4$

Step 2.8

Find the union of the solutions.

$x < - 4$ or $x > 4$

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 $0$ .

$x < - 4, x > 4$

$(- \infty, - 4) \cup (4, \infty)$

Step 4