Precalculus Examples

$f (x) = \frac{x - 1}{3 \sqrt{10 - 2 x}}$

Step 1

Set the radicand in $\sqrt{10 - 2 x}$ greater than or equal to $0$ to find where the expression is defined.

$10 - 2 x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $10$ from both sides of the inequality.

$- 2 x \geq - 10$

Step 2.2

Divide each term in $- 2 x \geq - 10$ by $- 2$ and simplify.

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Step 2.2.1

Divide each term in $- 2 x \geq - 10$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 10}{- 2}$

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.

$\frac{- 2 x}{- 2} \leq \frac{- 10}{- 2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 10}{- 2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 10$ by $- 2$ .

$x \leq 5$

Step 3

Set the denominator in $\frac{x - 1}{3 \sqrt{10 - 2 x}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt{10 - 2 x} = 0$

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.

${(3 \sqrt{10 - 2 x})}^{2} = 0^{2}$

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 $\sqrt{10 - 2 x}$ as ${(10 - 2 x)}^{\frac{1}{2}}$ .

${(3 {(10 - 2 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${(3 {(10 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Apply the product rule to $3 {(10 - 2 x)}^{\frac{1}{2}}$ .

$3^{2} {({(10 - 2 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2.1.2

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

$9 {({(10 - 2 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2.1.3

Multiply the exponents in ${({(10 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.3.1

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

$9 {(10 - 2 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.3.2.1

Cancel the common factor.

$9 {(10 - 2 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.3.2.2

Rewrite the expression.

$9 {(10 - 2 x)}^{1} = 0^{2}$

Step 4.2.2.1.4

Simplify.

$9 (10 - 2 x) = 0^{2}$

Step 4.2.2.1.5

Apply the distributive property.

$9 \cdot 10 + 9 (- 2 x) = 0^{2}$

Step 4.2.2.1.6

Multiply.

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Step 4.2.2.1.6.1

Multiply $9$ by $10$ .

$90 + 9 (- 2 x) = 0^{2}$

Step 4.2.2.1.6.2

Multiply $- 2$ by $9$ .

$90 - 18 x = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$90 - 18 x = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Subtract $90$ from both sides of the equation.

$- 18 x = - 90$

Step 4.3.2

Divide each term in $- 18 x = - 90$ by $- 18$ and simplify.

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Step 4.3.2.1

Divide each term in $- 18 x = - 90$ by $- 18$ .

$\frac{- 18 x}{- 18} = \frac{- 90}{- 18}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $- 18$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{- 18 x}{- 18} = \frac{- 90}{- 18}$

Step 4.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 90}{- 18}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Divide $- 90$ by $- 18$ .

$x = 5$

Step 5

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

Interval Notation:

$(- \infty, 5)$

Set-Builder Notation:

${x | x < 5}$

Step 6