Finite Math Examples

$\sqrt{2 x} (\sqrt{8 x - \sqrt{32}})$

Step 1

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

$2 x \geq 0$

Step 2

Divide each term in $2 x \geq 0$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 x \geq 0$ by $2$ .

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $2$ .

$x \geq 0$

Step 3

Set the radicand in $\sqrt{8 x - \sqrt{32}}$ greater than or equal to $0$ to find where the expression is defined.

$8 x - \sqrt{32} \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 4.1.1.1

Factor $16$ out of $32$ .

$8 x - \sqrt{16 (2)} \geq 0$

Step 4.1.1.2

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

$8 x - \sqrt{4^{2} \cdot 2} \geq 0$

Step 4.1.2

Pull terms out from under the radical.

$8 x - (4 \sqrt{2}) \geq 0$

Step 4.1.3

Multiply $4$ by $- 1$ .

$8 x - 4 \sqrt{2} \geq 0$

Step 4.2

Add $4 \sqrt{2}$ to both sides of the inequality.

$8 x \geq 4 \sqrt{2}$

Step 4.3

Divide each term in $8 x \geq 4 \sqrt{2}$ by $8$ and simplify.

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Step 4.3.1

Divide each term in $8 x \geq 4 \sqrt{2}$ by $8$ .

$\frac{8 x}{8} \geq \frac{4 \sqrt{2}}{8}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $8$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{4 \sqrt{2}}{8}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{4 \sqrt{2}}{8}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Cancel the common factor of $4$ and $8$ .

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Step 4.3.3.1.1

Factor $4$ out of $4 \sqrt{2}$ .

$x \geq \frac{4 (\sqrt{2})}{8}$

Step 4.3.3.1.2

Cancel the common factors.

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Step 4.3.3.1.2.1

Factor $4$ out of $8$ .

$x \geq \frac{4 \sqrt{2}}{4 \cdot 2}$

Step 4.3.3.1.2.2

Cancel the common factor.

$x \geq \frac{4 \sqrt{2}}{4 \cdot 2}$

Step 4.3.3.1.2.3

Rewrite the expression.

$x \geq \frac{\sqrt{2}}{2}$

Step 5

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

Interval Notation:

$[\frac{\sqrt{2}}{2}, \infty)$

Set-Builder Notation:

${x | x \geq \frac{\sqrt{2}}{2}}$

Step 6