Solve the Inequality for x 3 square root of x-4>=5

$3 \sqrt{x} - 4 \geq 5$

Step 1

Move all terms not containing $3 \sqrt{x}$ to the right side of the inequality.

Add $4$ to both sides of the inequality.

$3 \sqrt{x} \geq 5 + 4$

Add $5$ and $4$ .

$3 \sqrt{x} \geq 9$

Step 2

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

${(3 \sqrt{x})}^{2} \geq 9^{2}$

Step 3

Simplify each side of the inequality.

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

${(3 x^{\frac{1}{2}})}^{2} \geq 9^{2}$

Simplify the left side.

Simplify ${(3 x^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $3 x^{\frac{1}{2}}$ .

$3^{2} {(x^{\frac{1}{2}})}^{2} \geq 9^{2}$

Raise $3$ to the power of $2$ .

$9 {(x^{\frac{1}{2}})}^{2} \geq 9^{2}$

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

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

$9 x^{\frac{1}{2} \cdot 2} \geq 9^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 x^{\frac{1}{2} \cdot 2} \geq 9^{2}$

Rewrite the expression.

$9 x^{1} \geq 9^{2}$

Simplify.

$9 x \geq 9^{2}$

Simplify the right side.

Raise $9$ to the power of $2$ .

$9 x \geq 81$

Step 4

Divide each term in $9 x \geq 81$ by $9$ and simplify.

Divide each term in $9 x \geq 81$ by $9$ .

$\frac{9 x}{9} \geq \frac{81}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} \geq \frac{81}{9}$

Divide $x$ by $1$ .

$x \geq \frac{81}{9}$

Simplify the right side.

Divide $81$ by $9$ .

$x \geq 9$

Step 5

Find the domain of $3 \sqrt{x} - 9$ .

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

$x \geq 0$

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

$[0, \infty)$

Step 6

The solution consists of all of the true intervals.

$x \geq 9$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x \geq 9$

Interval Notation:

$[9, \infty)$

Step 8

Solve the Inequality for x 3 square root of x-4>=5

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