Precalculus Examples

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

Step 1

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

$x^{2} \geq 0$

Step 2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 3

Set the radicand in $\sqrt{\sqrt{x^{2}} - 1}$ greater than or equal to $0$ to find where the expression is defined.

$\sqrt{x^{2}} - 1 \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $1$ to both sides of the inequality.

$\sqrt{x^{2}} \geq 1$

Step 4.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{x^{2}}}^{2} \geq 1^{2}$

Step 4.3

Simplify each side of the inequality.

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Step 4.3.1

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

${(x^{\frac{2}{2}})}^{2} \geq 1^{2}$

Step 4.3.2

Divide $2$ by $2$ .

${(x^{1})}^{2} \geq 1^{2}$

Step 4.3.3

Simplify the left side.

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Step 4.3.3.1

Multiply the exponents in ${(x^{1})}^{2}$ .

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Step 4.3.3.1.1

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

$x^{1 \cdot 2} \geq 1^{2}$

Step 4.3.3.1.2

Multiply $2$ by $1$ .

$x^{2} \geq 1^{2}$

Step 4.3.4

Simplify the right side.

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Step 4.3.4.1

One to any power is one.

$x^{2} \geq 1$

Step 4.4

Solve for $x$ .

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Step 4.4.1

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

$\sqrt{x^{2}} \geq \sqrt{1}$

Step 4.4.2

Simplify the equation.

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Step 4.4.2.1

Simplify the left side.

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Step 4.4.2.1.1

Pull terms out from under the radical.

$| x | \geq \sqrt{1}$

Step 4.4.2.2

Simplify the right side.

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Step 4.4.2.2.1

Any root of $1$ is $1$ .

$| x | \geq 1$

Step 4.4.3

Write $| x | \geq 1$ as a piecewise.

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Step 4.4.3.1

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

$x \geq 0$

Step 4.4.3.2

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

$x \geq 1$

Step 4.4.3.3

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

$x < 0$

Step 4.4.3.4

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

$- x \geq 1$

Step 4.4.3.5

Write as a piecewise.

${\begin{cases} x \geq 1 & x \geq 0 \\ - x \geq 1 & x < 0 \end{cases}$

Step 4.4.4

Find the intersection of $x \geq 1$ and $x \geq 0$ .

$x \geq 1$

Step 4.4.5

Divide each term in $- x \geq 1$ by $- 1$ and simplify.

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Step 4.4.5.1

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

$\frac{- x}{- 1} \leq \frac{1}{- 1}$

Step 4.4.5.2

Simplify the left side.

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Step 4.4.5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{1}{- 1}$

Step 4.4.5.2.2

Divide $x$ by $1$ .

$x \leq \frac{1}{- 1}$

Step 4.4.5.3

Simplify the right side.

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Step 4.4.5.3.1

Divide $1$ by $- 1$ .

$x \leq - 1$

Step 4.4.6

Find the union of the solutions.

$x \leq - 1$ or $x \geq 1$

Step 5

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

Interval Notation:

$(- \infty, - 1] \cup [1, \infty)$

Set-Builder Notation:

${x | x \leq - 1, x \geq 1}$

Step 6