Precalculus Examples

$f (x) = \frac{\sqrt{2 x + 7}}{\sqrt{3 - 2 x}}$

Step 1

Set the radicand in $\sqrt{2 x + 7}$ greater than or equal to $0$ to find where the expression is defined.

$2 x + 7 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $7$ from both sides of the inequality.

$2 x \geq - 7$

Step 2.2

Divide each term in $2 x \geq - 7$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x \geq - 7$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 7}{2}$

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.

$\frac{2 x}{2} \geq \frac{- 7}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 7}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{7}{2}$

Step 3

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

$3 - 2 x \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $3$ from both sides of the inequality.

$- 2 x \geq - 3$

Step 4.2

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

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Step 4.2.1

Divide each term in $- 2 x \geq - 3$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 3}{- 2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{- 3}{- 2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 3}{- 2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Dividing two negative values results in a positive value.

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

Step 5

Set the denominator in $\frac{\sqrt{2 x + 7}}{\sqrt{3 - 2 x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{3 - 2 x} = 0$

Step 6

Solve for $x$ .

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Step 6.1

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

${\sqrt{3 - 2 x}}^{2} = 0^{2}$

Step 6.2

Simplify each side of the equation.

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Step 6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - 2 x}$ as ${(3 - 2 x)}^{\frac{1}{2}}$ .

${({(3 - 2 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

Multiply the exponents in ${({(3 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.2.1.1.1

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

${(3 - 2 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.2.1

Cancel the common factor.

${(3 - 2 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.2.1.1.2.2

Rewrite the expression.

${(3 - 2 x)}^{1} = 0^{2}$

Step 6.2.2.1.2

Simplify.

$3 - 2 x = 0^{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

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

$3 - 2 x = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Subtract $3$ from both sides of the equation.

$- 2 x = - 3$

Step 6.3.2

Divide each term in $- 2 x = - 3$ by $- 2$ and simplify.

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Step 6.3.2.1

Divide each term in $- 2 x = - 3$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 3}{- 2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $- 2$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 3}{- 2}$

Step 6.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Dividing two negative values results in a positive value.

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

Step 7

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

Interval Notation:

$[- \frac{7}{2}, \frac{3}{2})$

Set-Builder Notation:

${x | - \frac{7}{2} \leq x < \frac{3}{2}}$

Step 8