Solve n*(2/3)^3<n*(7/2)^2

$$n \cdot ( 2 / 3 ) ^ { 3 } < n \cdot ( 7 / 2 ) ^ { 2 }$$

Solve for n

$$n>0$$

Solution Steps

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

$$n\times \frac{8}{27}<n\times \left(\frac{7}{2}\right)^{2}$$

Calculate \frac{7}{2} to the power of 2 and get \frac{49}{4}.

$$n\times \frac{8}{27}<n\times \frac{49}{4}$$

Subtract n\times \frac{49}{4} from both sides.

$$n\times \frac{8}{27}-n\times \frac{49}{4}<0$$

Combine n\times \frac{8}{27} and -n\times \frac{49}{4} to get -\frac{1291}{108}n.

$$-\frac{1291}{108}n<0$$

Product of two numbers is <0 if one is >0 and the other is <0. Since -\frac{1291}{108}<0, n must be >0.

$$n>0$$