Algebra Examples

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

Step 1

Rewrite so $x$ is on the left side of the inequality.

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

Step 2

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

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Step 2.1

Apply the product rule to $\frac{x}{8}$ .

$\frac{x^{\frac{1}{3}}}{8^{\frac{1}{3}}} < \frac{1}{2}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Rewrite $8$ as $2^{3}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Cancel the common factor of $3$ .

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Step 2.2.3.1

Cancel the common factor.

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

Step 2.2.3.2

Rewrite the expression.

$\frac{x^{\frac{1}{3}}}{2^{1}} < \frac{1}{2}$

Step 2.2.4

Evaluate the exponent.

$\frac{x^{\frac{1}{3}}}{2} < \frac{1}{2}$

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

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

Step 4

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 1$

Step 5

The solution consists of all of the true intervals.

$x < 1$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x < 1$

Interval Notation:

$(- \infty, 1)$

Step 7