Precalculus Examples

$\frac{1}{\sqrt{2 x - 1}} + \frac{3}{x^{2} - 4}$

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 denominator in $\frac{1}{\sqrt{2 x - 1}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{2 x - 1} = 0$

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

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

${({(2 x - 1)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

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

${(2 x - 1)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(2 x - 1)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(2 x - 1)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$2 x - 1 = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raising $0$ to any positive power yields $0$ .

$2 x - 1 = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 4.3.2

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

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Step 4.3.2.1

Divide each term in $2 x = 1$ by $2$ .

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

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

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

Step 4.3.2.2.1.2

Divide $x$ by $1$ .

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

Step 5

Set the denominator in $\frac{3}{x^{2} - 4}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 = 0$

Step 6

Solve for $x$ .

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Step 6.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 6.2

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

$x = \pm \sqrt{4}$

Step 6.3

Simplify $\pm \sqrt{4}$ .

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Step 6.3.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 6.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 6.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 6.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 6.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 7

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

Interval Notation:

$(\frac{1}{2}, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x > \frac{1}{2}, x \neq 2}$

Step 8