Solve n{left(2/3right)}^3<n{left(1/2right)}^2

$$n { \left( \frac{ 2 }{ 3 } \right) }^{ 3 } < n { \left( \frac{ 1 }{ 2 } \right) }^{ 2 }$$

Solve for n

$$n<0$$

Solution Steps

Calculate \frac{2}{3} to the power of 3 and get \frac{8}{27}.

$$n\times \frac{8}{27}<n\times \left(\frac{1}{2}\right)^{2}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$n\times \frac{8}{27}<n\times \frac{1}{4}$$

Subtract n\times \frac{1}{4} from both sides.

$$n\times \frac{8}{27}-n\times \frac{1}{4}<0$$

Combine n\times \frac{8}{27} and -n\times \frac{1}{4} to get \frac{5}{108}n.

$$\frac{5}{108}n<0$$

Product of two numbers is <0 if one is >0 and the other is <0. Since \frac{5}{108}>0, n must be <0.

$$n<0$$