Precalculus Examples

$\sqrt{8 - 2 x} - \sqrt{5 x - 12}$

Step 1

Set the radicand in $\sqrt{8 - 2 x}$ greater than or equal to $0$ to find where the expression is defined.

$8 - 2 x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $8$ from both sides of the inequality.

$- 2 x \geq - 8$

Step 2.2

Divide each term in $- 2 x \geq - 8$ by $- 2$ and simplify.

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Step 2.2.1

Divide each term in $- 2 x \geq - 8$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 8}{- 2}$

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.

$\frac{- 2 x}{- 2} \leq \frac{- 8}{- 2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 8}{- 2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 8$ by $- 2$ .

$x \leq 4$

Step 3

Set the radicand in $\sqrt{5 x - 12}$ greater than or equal to $0$ to find where the expression is defined.

$5 x - 12 \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $12$ to both sides of the inequality.

$5 x \geq 12$

Step 4.2

Divide each term in $5 x \geq 12$ by $5$ and simplify.

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Step 4.2.1

Divide each term in $5 x \geq 12$ by $5$ .

$\frac{5 x}{5} \geq \frac{12}{5}$

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.

$\frac{5 x}{5} \geq \frac{12}{5}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{12}{5}$

Step 5

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

Interval Notation:

$[\frac{12}{5}, 4]$

Set-Builder Notation:

${x | \frac{12}{5} \leq x \leq 4}$

Step 6