Precalculus Examples

$\sqrt{| x | + 1}$

Step 1

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

$| x | + 1 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Write $| x | + 1 \geq 0$ as a piecewise.

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Step 2.1.1

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

$x \geq 0$

Step 2.1.2

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

$x + 1 \geq 0$

Step 2.1.3

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

$x < 0$

Step 2.1.4

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

$- x + 1 \geq 0$

Step 2.1.5

Write as a piecewise.

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

Step 2.2

Solve $x + 1 \geq 0$ when $x \geq 0$ .

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Step 2.2.1

Subtract $1$ from both sides of the inequality.

$x \geq - 1$

Step 2.2.2

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

$x \geq 0$

Step 2.3

Solve $- x + 1 \geq 0$ when $x < 0$ .

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Step 2.3.1

Solve $- x + 1 \geq 0$ for $x$ .

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Step 2.3.1.1

Subtract $1$ from both sides of the inequality.

$- x \geq - 1$

Step 2.3.1.2

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

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Step 2.3.1.2.1

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 2.3.1.2.2

Simplify the left side.

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Step 2.3.1.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.1.2.2.2

Divide $x$ by $1$ .

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

Step 2.3.1.2.3

Simplify the right side.

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Step 2.3.1.2.3.1

Divide $- 1$ by $- 1$ .

$x \leq 1$

Step 2.3.2

Find the intersection of $x \leq 1$ and $x < 0$ .

$x < 0$

Step 2.4

Find the union of the solutions.

All real numbers

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4