Precalculus Examples

$f (x) = \sqrt{\frac{4}{5} - 2 x}$

Step 1

Set the radicand in $\sqrt{\frac{4}{5} - 2 x}$ greater than or equal to $0$ to find where the expression is defined.

$\frac{4}{5} - 2 x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $\frac{4}{5}$ from both sides of the inequality.

$- 2 x \geq - \frac{4}{5}$

Step 2.2

Divide each term in $- 2 x \geq - \frac{4}{5}$ by $- 2$ and simplify.

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Step 2.2.1

Divide each term in $- 2 x \geq - \frac{4}{5}$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- \frac{4}{5}}{- 2}$

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.

$\frac{- 2 x}{- 2} \leq \frac{- \frac{4}{5}}{- 2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- \frac{4}{5}}{- 2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x \leq - \frac{4}{5} \cdot \frac{1}{- 2}$

Step 2.2.3.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.1

Move the leading negative in $- \frac{4}{5}$ into the numerator.

$x \leq \frac{- 4}{5} \cdot \frac{1}{- 2}$

Step 2.2.3.2.2

Factor $2$ out of $- 4$ .

$x \leq \frac{2 (- 2)}{5} \cdot \frac{1}{- 2}$

Step 2.2.3.2.3

Factor $2$ out of $- 2$ .

$x \leq \frac{2 \cdot - 2}{5} \cdot \frac{1}{2 \cdot - 1}$

Step 2.2.3.2.4

Cancel the common factor.

$x \leq \frac{2 \cdot - 2}{5} \cdot \frac{1}{2 \cdot - 1}$

Step 2.2.3.2.5

Rewrite the expression.

$x \leq \frac{- 2}{5} \cdot \frac{1}{- 1}$

Step 2.2.3.3

Multiply $\frac{- 2}{5}$ by $\frac{1}{- 1}$ .

$x \leq \frac{- 2}{5 \cdot - 1}$

Step 2.2.3.4

Multiply $5$ by $- 1$ .

$x \leq \frac{- 2}{- 5}$

Step 2.2.3.5

Dividing two negative values results in a positive value.

$x \leq \frac{2}{5}$

Step 3

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

Interval Notation:

$(- \infty, \frac{2}{5}]$

Set-Builder Notation:

${x | x \leq \frac{2}{5}}$

Step 4