Algebra Examples

$\sqrt{2 x + 4} + 1 \geq 5$

Step 1

Move all terms not containing $\sqrt{2 x + 4}$ to the right side of the inequality.

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Step 1.1

Subtract $1$ from both sides of the inequality.

$\sqrt{2 x + 4} \geq 5 - 1$

Step 1.2

Subtract $1$ from $5$ .

$\sqrt{2 x + 4} \geq 4$

Step 2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{2 x + 4}}^{2} \geq 4^{2}$

Step 3

Simplify each side of the inequality.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 4}$ as ${(2 x + 4)}^{\frac{1}{2}}$ .

${({(2 x + 4)}^{\frac{1}{2}})}^{2} \geq 4^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(2 x + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(2 x + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x + 4)}^{\frac{1}{2} \cdot 2} \geq 4^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(2 x + 4)}^{\frac{1}{2} \cdot 2} \geq 4^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x + 4)}^{1} \geq 4^{2}$

Step 3.2.1.2

Simplify.

$2 x + 4 \geq 4^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $4$ to the power of $2$ .

$2 x + 4 \geq 16$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.1.1

Subtract $4$ from both sides of the inequality.

$2 x \geq 16 - 4$

Step 4.1.2

Subtract $4$ from $16$ .

$2 x \geq 12$

Step 4.2

Divide each term in $2 x \geq 12$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 x \geq 12$ by $2$ .

$\frac{2 x}{2} \geq \frac{12}{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{12}{2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{12}{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $12$ by $2$ .

$x \geq 6$

Step 5

Find the domain of $\sqrt{2 (x + 2)} - 4$ .

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Step 5.1

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 5.2

Solve for $x$ .

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Step 5.2.1

Divide each term in $2 (x + 2) \geq 0$ by $2$ and simplify.

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Step 5.2.1.1

Divide each term in $2 (x + 2) \geq 0$ by $2$ .

$\frac{2 (x + 2)}{2} \geq \frac{0}{2}$

Step 5.2.1.2

Simplify the left side.

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Step 5.2.1.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.2.1.1

Cancel the common factor.

$\frac{2 (x + 2)}{2} \geq \frac{0}{2}$

Step 5.2.1.2.1.2

Divide $x + 2$ by $1$ .

$x + 2 \geq \frac{0}{2}$

Step 5.2.1.3

Simplify the right side.

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Step 5.2.1.3.1

Divide $0$ by $2$ .

$x + 2 \geq 0$

Step 5.2.2

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 5.3

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

$[- 2, \infty)$

Step 6

The solution consists of all of the true intervals.

$x \geq 6$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x \geq 6$

Interval Notation:

$[6, \infty)$

Step 8