Solve the Inequality for x 4 square root of x-2>20

$4 \sqrt{x - 2} > 20$

Step 1

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

${(4 \sqrt{x - 2})}^{2} > 20^{2}$

Step 2

Simplify each side of the inequality.

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

${(4 {(x - 2)}^{\frac{1}{2}})}^{2} > 20^{2}$

Simplify the left side.

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

Apply the product rule to $4 {(x - 2)}^{\frac{1}{2}}$ .

$4^{2} {({(x - 2)}^{\frac{1}{2}})}^{2} > 20^{2}$

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

$16 {({(x - 2)}^{\frac{1}{2}})}^{2} > 20^{2}$

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

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

$16 {(x - 2)}^{\frac{1}{2} \cdot 2} > 20^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x - 2)}^{\frac{1}{2} \cdot 2} > 20^{2}$

Rewrite the expression.

$16 {(x - 2)}^{1} > 20^{2}$

Simplify.

$16 (x - 2) > 20^{2}$

Apply the distributive property.

$16 x + 16 \cdot - 2 > 20^{2}$

Multiply $16$ by $- 2$ .

$16 x - 32 > 20^{2}$

Simplify the right side.

Raise $20$ to the power of $2$ .

$16 x - 32 > 400$

Step 3

Solve for $x$ .

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

Add $32$ to both sides of the inequality.

$16 x > 400 + 32$

Add $400$ and $32$ .

$16 x > 432$

Divide each term in $16 x > 432$ by $16$ and simplify.

Divide each term in $16 x > 432$ by $16$ .

$\frac{16 x}{16} > \frac{432}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} > \frac{432}{16}$

Divide $x$ by $1$ .

$x > \frac{432}{16}$

Simplify the right side.

Divide $432$ by $16$ .

$x > 27$

Step 4

Find the domain of $4 \sqrt{x - 2} - 20$ .

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

$x - 2 \geq 0$

Add $2$ to both sides of the inequality.

$x \geq 2$

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

$[2, \infty)$

Step 5

The solution consists of all of the true intervals.

$x > 27$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x > 27$

Interval Notation:

$(27, \infty)$

Step 7

Solve the Inequality for x 4 square root of x-2>20

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