Precalculus Examples

$f (x) = \sqrt{11 - \frac{2}{3} x}$

Step 1

Set the radicand in $\sqrt{11 - \frac{2}{3} x}$ greater than or equal to $0$ to find where the expression is defined.

$11 - \frac{2}{3} x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $x$ and $\frac{2}{3}$ .

$11 - \frac{x \cdot 2}{3} \geq 0$

Step 2.1.2

Move $2$ to the left of $x$ .

$11 - \frac{2 x}{3} \geq 0$

Step 2.2

Subtract $11$ from both sides of the inequality.

$- \frac{2 x}{3} \geq - 11$

Step 2.3

Divide each term in $- \frac{2 x}{3} \geq - 11$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- \frac{2 x}{3} \geq - 11$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{2 x}{3}}{- 1} \leq \frac{- 11}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{2 x}{3}}{1} \leq \frac{- 11}{- 1}$

Step 2.3.2.2

Divide $\frac{2 x}{3}$ by $1$ .

$\frac{2 x}{3} \leq \frac{- 11}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 11$ by $- 1$ .

$\frac{2 x}{3} \leq 11$

Step 2.4

Multiply both sides by $3$ .

$\frac{2 x}{3} \cdot 3 \leq 11 \cdot 3$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Cancel the common factor of $3$ .

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Step 2.5.1.1.1

Cancel the common factor.

$\frac{2 x}{3} \cdot 3 \leq 11 \cdot 3$

Step 2.5.1.1.2

Rewrite the expression.

$2 x \leq 11 \cdot 3$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Multiply $11$ by $3$ .

$2 x \leq 33$

Step 2.6

Divide each term in $2 x \leq 33$ by $2$ and simplify.

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Step 2.6.1

Divide each term in $2 x \leq 33$ by $2$ .

$\frac{2 x}{2} \leq \frac{33}{2}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $2$ .

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Step 2.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{33}{2}$

Step 2.6.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{33}{2}$

Step 3

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

Interval Notation:

$(- \infty, \frac{33}{2}]$

Set-Builder Notation:

${x | x \leq \frac{33}{2}}$

Step 4