Solve the Inequality for x square root of 2x+2+1>=5

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

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

Subtract $1$ from both sides of the inequality.

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

Subtract $1$ from $5$ .

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

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

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

Simplify each side of the inequality.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$2 x + 2 \geq 16$

Solve for $x$ .

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

Subtract $2$ from both sides of the inequality.

$2 x \geq 16 - 2$

Subtract $2$ from $16$ .

$2 x \geq 14$

Divide each term in $2 x \geq 14$ by $2$ and simplify.

Divide each term in $2 x \geq 14$ by $2$ .

$\frac{2 x}{2} \geq \frac{14}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{14}{2}$

Divide $x$ by $1$ .

$x \geq \frac{14}{2}$

Simplify the right side.

Divide $14$ by $2$ .

$x \geq 7$

Find the domain of $\sqrt{2 (x + 1)} - 4$ .

Set the radicand in $\sqrt{2 (x + 1)}$ greater than or equal to $0$ to find where the expression is defined.

$2 (x + 1) \geq 0$

Solve for $x$ .

Divide each term in $2 (x + 1) \geq 0$ by $2$ and simplify.

Divide each term in $2 (x + 1) \geq 0$ by $2$ .

$\frac{2 (x + 1)}{2} \geq \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (x + 1)}{2} \geq \frac{0}{2}$

Divide $x + 1$ by $1$ .

$x + 1 \geq \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x + 1 \geq 0$

Subtract $1$ from both sides of the inequality.

$x \geq - 1$

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

$[- 1, \infty)$

The solution consists of all of the true intervals.

$x \geq 7$

The result can be shown in multiple forms.

Inequality Form:

$x \geq 7$

Interval Notation:

$[7, \infty)$

Solve the Inequality for x square root of 2x+2+1>=5

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