Algebra Examples

$\sqrt[3]{x + 4} > \sqrt[3]{- x}$

Step 1

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

${\sqrt[3]{x + 4}}^{3} > {\sqrt[3]{- x}}^{3}$

Step 2

Simplify each side of the inequality.

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Step 2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{x + 4}$ as

${(x + 4)}^{\frac{1}{3}}$ .

${({(x + 4)}^{\frac{1}{3}})}^{3} > {\sqrt[3]{- x}}^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify

${({(x + 4)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in

${({(x + 4)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

${(x + 4)}^{\frac{1}{3} \cdot 3} > {\sqrt[3]{- x}}^{3}$

Step 2.2.1.1.2

Cancel the common factor of

$3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x + 4)}^{\frac{1}{3} \cdot 3} > {\sqrt[3]{- x}}^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x + 4)}^{1} > {\sqrt[3]{- x}}^{3}$

${(x + 4)}^{1} > {\sqrt[3]{- x}}^{3}$

${(x + 4)}^{1} > {\sqrt[3]{- x}}^{3}$

Step 2.2.1.2

Simplify.

$x + 4 > {\sqrt[3]{- x}}^{3}$

$x + 4 > {\sqrt[3]{- x}}^{3}$

$x + 4 > {\sqrt[3]{- x}}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify

${\sqrt[3]{- x}}^{3}$ .

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Step 2.3.1.1

Rewrite

$- x$ as

${({(- 1)}^{3})}^{3} x$ .

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Step 2.3.1.1.1

Rewrite

$- 1$ as

${(- 1)}^{3}$ .

$x + 4 > {\sqrt[3]{{(- 1)}^{3} x}}^{3}$

Step 2.3.1.1.2

Rewrite

$- 1$ as

${(- 1)}^{3}$ .

$x + 4 > {\sqrt[3]{{({(- 1)}^{3})}^{3} x}}^{3}$

$x + 4 > {\sqrt[3]{{({(- 1)}^{3})}^{3} x}}^{3}$

Step 2.3.1.2

Pull terms out from under the radical.

$x + 4 > {({(- 1)}^{3} \sqrt[3]{x})}^{3}$

Step 2.3.1.3

Simplify the expression.

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Step 2.3.1.3.1

Raise

$- 1$ to the power of

$3$ .

$x + 4 > {(- \sqrt[3]{x})}^{3}$

Step 2.3.1.3.2

Apply the product rule to

$- \sqrt[3]{x}$ .

$x + 4 > {(- 1)}^{3} {\sqrt[3]{x}}^{3}$

Step 2.3.1.3.3

Raise

$- 1$ to the power of

$3$ .

$x + 4 > - {\sqrt[3]{x}}^{3}$

$x + 4 > - {\sqrt[3]{x}}^{3}$

Step 2.3.1.4

Rewrite

${\sqrt[3]{x}}^{3}$ as

$x$ .

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Step 2.3.1.4.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{x}$ as

$x^{\frac{1}{3}}$ .

$x + 4 > - {(x^{\frac{1}{3}})}^{3}$

Step 2.3.1.4.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$x + 4 > - x^{\frac{1}{3} \cdot 3}$

Step 2.3.1.4.3

Combine

$\frac{1}{3}$ and

$3$ .

$x + 4 > - x^{\frac{3}{3}}$

Step 2.3.1.4.4

Cancel the common factor of

$3$ .

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Step 2.3.1.4.4.1

Cancel the common factor.

$x + 4 > - x^{\frac{3}{3}}$

Step 2.3.1.4.4.2

Rewrite the expression.

$x + 4 > - x^{1}$

$x + 4 > - x^{1}$

Step 2.3.1.4.5

Simplify.

$x + 4 > - x$

$x + 4 > - x$

$x + 4 > - x$

$x + 4 > - x$

$x + 4 > - x$

Step 3

Solve for

$x$ .

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Step 3.1

Move all terms containing

$x$ to the left side of the inequality.

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Step 3.1.1

Add

$x$ to both sides of the inequality.

$x + 4 + x > 0$

Step 3.1.2

Add

$x$ and

$x$ .

$2 x + 4 > 0$

$2 x + 4 > 0$

Step 3.2

Subtract

$4$ from both sides of the inequality.

$2 x > - 4$

Step 3.3

Divide each term in

$2 x > - 4$ by

$2$ and simplify.

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Step 3.3.1

Divide each term in

$2 x > - 4$ by

$2$ .

$\frac{2 x}{2} > \frac{- 4}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of

$2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} > \frac{- 4}{2}$

Step 3.3.2.1.2

Divide

$x$ by

$1$ .

$x > \frac{- 4}{2}$

$x > \frac{- 4}{2}$

$x > \frac{- 4}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide

$- 4$ by

$2$ .

$x > - 2$

$x > - 2$

$x > - 2$

$x > - 2$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x > - 2$

Interval Notation:

$(- 2, \infty)$

Step 5

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$