Linear Algebra Examples

$\sqrt{22 x + 33} = x + 7$

Step 1

Set the radicand in $\sqrt{22 x + 33}$ greater than or equal to $0$ to find where the expression is defined.

$22 x + 33 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $33$ from both sides of the inequality.

$22 x \geq - 33$

Step 2.2

Divide each term in $22 x \geq - 33$ by $22$ and simplify.

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Step 2.2.1

Divide each term in $22 x \geq - 33$ by $22$ .

$\frac{22 x}{22} \geq \frac{- 33}{22}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $22$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{22 x}{22} \geq \frac{- 33}{22}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 33}{22}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $- 33$ and $22$ .

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Step 2.2.3.1.1

Factor $11$ out of $- 33$ .

$x \geq \frac{11 (- 3)}{22}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $11$ out of $22$ .

$x \geq \frac{11 \cdot - 3}{11 \cdot 2}$

Step 2.2.3.1.2.2

Cancel the common factor.

$x \geq \frac{11 \cdot - 3}{11 \cdot 2}$

Step 2.2.3.1.2.3

Rewrite the expression.

$x \geq \frac{- 3}{2}$

Step 2.2.3.2

Move the negative in front of the fraction.

$x \geq - \frac{3}{2}$

Step 3

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

Interval Notation:

$[- \frac{3}{2}, \infty)$

Set-Builder Notation:

${x | x \geq - \frac{3}{2}}$

Step 4