Algebra Examples

$2 \sqrt[3]{x} - 5 \geq 3$

Step 1

Move all terms not containing $2 \sqrt[3]{x}$ to the right side of the inequality.

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Step 1.1

Add $5$ to both sides of the inequality.

$2 \sqrt[3]{x} \geq 3 + 5$

Step 1.2

Add $3$ and $5$ .

$2 \sqrt[3]{x} \geq 8$

Step 2

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

${(2 \sqrt[3]{x})}^{3} \geq 8^{3}$

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 $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

${(2 x^{\frac{1}{3}})}^{3} \geq 8^{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

$2^{3} {(x^{\frac{1}{3}})}^{3} \geq 8^{3}$

Step 3.2.1.2

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

$8 {(x^{\frac{1}{3}})}^{3} \geq 8^{3}$

Step 3.2.1.3

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 3.2.1.3.1

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

$8 x^{\frac{1}{3} \cdot 3} \geq 8^{3}$

Step 3.2.1.3.2

Cancel the common factor of $3$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$8 x^{\frac{1}{3} \cdot 3} \geq 8^{3}$

Step 3.2.1.3.2.2

Rewrite the expression.

$8 x^{1} \geq 8^{3}$

Step 3.2.1.4

Simplify.

$8 x \geq 8^{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $8$ to the power of $3$ .

$8 x \geq 512$

Step 4

Divide each term in $8 x \geq 512$ by $8$ and simplify.

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Step 4.1

Divide each term in $8 x \geq 512$ by $8$ .

$\frac{8 x}{8} \geq \frac{512}{8}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $8$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{512}{8}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{512}{8}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $512$ by $8$ .

$x \geq 64$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \geq 64$

Interval Notation:

$[64, \infty)$

Step 6