Precalculus Examples

$\frac{5}{\sqrt{x - 1} - 3}$

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

Add $1$ to both sides of the inequality.

$x \geq 1$

Step 3

Set the denominator in $\frac{5}{\sqrt{x - 1} - 3}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x - 1} - 3 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $3$ to both sides of the equation.

$\sqrt{x - 1} = 3$

Step 4.2

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

${\sqrt{x - 1}}^{2} = 3^{2}$

Step 4.3

Simplify each side of the equation.

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Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 1}$ as ${(x - 1)}^{\frac{1}{2}}$ .

${({(x - 1)}^{\frac{1}{2}})}^{2} = 3^{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

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

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Step 4.3.2.1.1

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

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Step 4.3.2.1.1.1

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

${(x - 1)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 4.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.1.2.1

Cancel the common factor.

${(x - 1)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 4.3.2.1.1.2.2

Rewrite the expression.

${(x - 1)}^{1} = 3^{2}$

Step 4.3.2.1.2

Simplify.

$x - 1 = 3^{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Raise $3$ to the power of $2$ .

$x - 1 = 9$

Step 4.4

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

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Step 4.4.1

Add $1$ to both sides of the equation.

$x = 9 + 1$

Step 4.4.2

Add $9$ and $1$ .

$x = 10$

Step 5

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

Interval Notation:

$[1, 10) \cup (10, \infty)$

Set-Builder Notation:

${x | x \geq 1, x \neq 10}$

Step 6