Precalculus Examples

$\sqrt{2 x + 9} + \sqrt{3 - 5 x}$

Step 1

Set the radicand in $\sqrt{2 x + 9}$ greater than or equal to $0$ to find where the expression is defined.

$2 x + 9 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $9$ from both sides of the inequality.

$2 x \geq - 9$

Step 2.2

Divide each term in $2 x \geq - 9$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x \geq - 9$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 9}{2}$

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.

$\frac{2 x}{2} \geq \frac{- 9}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 9}{2}$

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.

$x \geq - \frac{9}{2}$

Step 3

Set the radicand in $\sqrt{3 - 5 x}$ greater than or equal to $0$ to find where the expression is defined.

$3 - 5 x \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $3$ from both sides of the inequality.

$- 5 x \geq - 3$

Step 4.2

Divide each term in $- 5 x \geq - 3$ by $- 5$ and simplify.

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Step 4.2.1

Divide each term in $- 5 x \geq - 3$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} \leq \frac{- 3}{- 5}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 5$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} \leq \frac{- 3}{- 5}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 3}{- 5}$

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.

$x \leq \frac{3}{5}$

Step 5

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

Interval Notation:

$[- \frac{9}{2}, \frac{3}{5}]$

Set-Builder Notation:

${x | - \frac{9}{2} \leq x \leq \frac{3}{5}}$

Step 6