Precalculus Examples

$\sqrt{6 - x} - \sqrt{3 x - 9}$

Step 1

Set the radicand in $\sqrt{6 - x}$ greater than or equal to $0$ to find where the expression is defined.

$6 - x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $6$ from both sides of the inequality.

$- x \geq - 6$

Step 2.2

Divide each term in $- x \geq - 6$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- x \geq - 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 6}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 6}{- 1}$

Step 2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 6}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 6$ by $- 1$ .

$x \leq 6$

Step 3

Set the radicand in $\sqrt{3 x - 9}$ greater than or equal to $0$ to find where the expression is defined.

$3 x - 9 \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $9$ to both sides of the inequality.

$3 x \geq 9$

Step 4.2

Divide each term in $3 x \geq 9$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 x \geq 9$ by $3$ .

$\frac{3 x}{3} \geq \frac{9}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{9}{3}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{9}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $9$ by $3$ .

$x \geq 3$

Step 5

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

Interval Notation:

$[3, 6]$

Set-Builder Notation:

${x | 3 \leq x \leq 6}$

Step 6