Precalculus Examples

$g (x) = \sqrt{5 - \sqrt{2 x - 1}}$

Step 1

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

$2 x - 1 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $1$ to both sides of the inequality.

$2 x \geq 1$

Step 2.2

Divide each term in $2 x \geq 1$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x \geq 1$ by $2$ .

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

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.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3

Set the radicand in $\sqrt{5 - \sqrt{2 x - 1}}$ greater than or equal to $0$ to find where the expression is defined.

$5 - \sqrt{2 x - 1} \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $5$ from both sides of the inequality.

$- \sqrt{2 x - 1} \geq - 5$

Step 4.2

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

${(- \sqrt{2 x - 1})}^{2} \geq {(- 5)}^{2}$

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 $\sqrt{2 x - 1}$ as ${(2 x - 1)}^{\frac{1}{2}}$ .

${(- {(2 x - 1)}^{\frac{1}{2}})}^{2} \geq {(- 5)}^{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Simplify ${(- {(2 x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.1.1

Apply the product rule to $- {(2 x - 1)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(2 x - 1)}^{\frac{1}{2}})}^{2} \geq {(- 5)}^{2}$

Step 4.3.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(2 x - 1)}^{\frac{1}{2}})}^{2} \geq {(- 5)}^{2}$

Step 4.3.2.1.3

Multiply ${({(2 x - 1)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(2 x - 1)}^{\frac{1}{2}})}^{2} \geq {(- 5)}^{2}$

Step 4.3.2.1.4

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

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Step 4.3.2.1.4.1

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

${(2 x - 1)}^{\frac{1}{2} \cdot 2} \geq {(- 5)}^{2}$

Step 4.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.4.2.1

Cancel the common factor.

${(2 x - 1)}^{\frac{1}{2} \cdot 2} \geq {(- 5)}^{2}$

Step 4.3.2.1.4.2.2

Rewrite the expression.

${(2 x - 1)}^{1} \geq {(- 5)}^{2}$

Step 4.3.2.1.5

Simplify.

$2 x - 1 \geq {(- 5)}^{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Raise $- 5$ to the power of $2$ .

$2 x - 1 \geq 25$

Step 4.4

Solve for $x$ .

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Step 4.4.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.4.1.1

Add $1$ to both sides of the inequality.

$2 x \geq 25 + 1$

Step 4.4.1.2

Add $25$ and $1$ .

$2 x \geq 26$

Step 4.4.2

Divide each term in $2 x \geq 26$ by $2$ and simplify.

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Step 4.4.2.1

Divide each term in $2 x \geq 26$ by $2$ .

$\frac{2 x}{2} \geq \frac{26}{2}$

Step 4.4.2.2

Simplify the left side.

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Step 4.4.2.2.1

Cancel the common factor of $2$ .

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Step 4.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{26}{2}$

Step 4.4.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{26}{2}$

Step 4.4.2.3

Simplify the right side.

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Step 4.4.2.3.1

Divide $26$ by $2$ .

$x \geq 13$

Step 4.5

Find the domain of $5 - \sqrt{2 x - 1}$ .

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Step 4.5.1

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

$2 x - 1 \geq 0$

Step 4.5.2

Solve for $x$ .

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Step 4.5.2.1

Add $1$ to both sides of the inequality.

$2 x \geq 1$

Step 4.5.2.2

Divide each term in $2 x \geq 1$ by $2$ and simplify.

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Step 4.5.2.2.1

Divide each term in $2 x \geq 1$ by $2$ .

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

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

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

Step 4.5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.5.3

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

$[\frac{1}{2}, \infty)$

Step 4.6

Use each root to create test intervals.

$x < \frac{1}{2}$

$\frac{1}{2} < x < 13$

$x > 13$

Step 4.7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 4.7.1

Test a value on the interval $x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 4.7.1.1

Choose a value on the interval $x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 4.7.1.2

Replace $x$ with $0$ in the original inequality.

$5 - \sqrt{2 (0) - 1} \geq 0$

Step 4.7.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 4.7.2

Test a value on the interval $\frac{1}{2} < x < 13$ to see if it makes the inequality true.

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Step 4.7.2.1

Choose a value on the interval $\frac{1}{2} < x < 13$ and see if this value makes the original inequality true.

$x = 3$

Step 4.7.2.2

Replace $x$ with $3$ in the original inequality.

$5 - \sqrt{2 (3) - 1} \geq 0$

Step 4.7.2.3

The left side $2.76393202$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 4.7.3

Test a value on the interval $x > 13$ to see if it makes the inequality true.

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Step 4.7.3.1

Choose a value on the interval $x > 13$ and see if this value makes the original inequality true.

$x = 16$

Step 4.7.3.2

Replace $x$ with $16$ in the original inequality.

$5 - \sqrt{2 (16) - 1} \geq 0$

Step 4.7.3.3

The left side $- 0.56776436$ is less than the right side $0$ , which means that the given statement is false.

False

Step 4.7.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < 13$ True

$x > 13$ False

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < 13$ True

$x > 13$ False

Step 4.8

The solution consists of all of the true intervals.

$\frac{1}{2} \leq x \leq 13$

Step 5

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

Interval Notation:

$[\frac{1}{2}, 13]$

Set-Builder Notation:

${x | \frac{1}{2} \leq x \leq 13}$

Step 6