Precalculus Examples

$\sqrt{2 x + 2} + \sqrt{6 - 4 x}$

Step 1

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

$2 x + 2 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $2$ from both sides of the inequality.

$2 x \geq - 2$

Step 2.2

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

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Step 2.2.1

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

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

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.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 2$ by $2$ .

$x \geq - 1$

Step 3

Set the radicand in $\sqrt{6 - 4 x}$ greater than or equal to $0$ to find where the expression is defined.

$6 - 4 x \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $6$ from both sides of the inequality.

$- 4 x \geq - 6$

Step 4.2

Divide each term in $- 4 x \geq - 6$ by $- 4$ and simplify.

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Step 4.2.1

Divide each term in $- 4 x \geq - 6$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \leq \frac{- 6}{- 4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} \leq \frac{- 6}{- 4}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 6}{- 4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $- 6$ and $- 4$ .

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Step 4.2.3.1.1

Factor $- 2$ out of $- 6$ .

$x \leq \frac{- 2 (3)}{- 4}$

Step 4.2.3.1.2

Cancel the common factors.

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Step 4.2.3.1.2.1

Factor $- 2$ out of $- 4$ .

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

Step 4.2.3.1.2.2

Cancel the common factor.

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

Step 4.2.3.1.2.3

Rewrite the expression.

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

Step 5

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

Interval Notation:

$[- 1, \frac{3}{2}]$

Set-Builder Notation:

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

Step 6