Precalculus Examples

$h (x) = \frac{\sqrt{2 x - 3}}{x - 5}$

Step 1

Set the radicand in $\sqrt{2 x - 3}$ greater than or equal to $0$ to find where the expression is defined.

$2 x - 3 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $3$ to both sides of the inequality.

$2 x \geq 3$

Step 2.2

Divide each term in $2 x \geq 3$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x \geq 3$ by $2$ .

$\frac{2 x}{2} \geq \frac{3}{2}$

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.

$\frac{2 x}{2} \geq \frac{3}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{3}{2}$

Step 3

Set the denominator in $\frac{\sqrt{2 x - 3}}{x - 5}$ equal to $0$ to find where the expression is undefined.

$x - 5 = 0$

Step 4

Add $5$ to both sides of the equation.

$x = 5$

Step 5

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

Interval Notation:

$[\frac{3}{2}, 5) \cup (5, \infty)$

Set-Builder Notation:

${x | x \geq \frac{3}{2}, x \neq 5}$

Step 6