Question

Solve the inequality ${16}^n<8^{n+1}$. Check your solution.

Short answer

The solution for the inequality is $n<3$.

Step-by-step solution

Step 1. Write down the given inequality.

The given inequality is ${16}^n<8^{n+1}$.

Step 2. Simplify the inequality.

The given inequality ${16}^n<8^{n+1}$is solved as:

$\begin{array}{l}{16}^n<8^{n+1}\mathrm{....}\left(\text{Given}\right)\\ {\left(2^4\right)}^n<{\left(2^3\right)}^{n+1}\\ {\left(2\right)}^{4n}<{\left(2\right)}^{3n+3}\\ 4n<3n+3\\ n<3\end{array}$

Step 3. Verify the result for obtained value.

Plugging $n=2$in given inequality and verify the result.

$\begin{array}{l}{16}^n<8^{n+1}\mathrm{....}\left(\text{Given}\right)\\ {16}^{\left(2\right)}<8^{\left(2\right)+1}\\ {16}^2<8^3\\ 256<\mathrm{512....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $n=2$.

Step 4. Conclusion.

The solution for the given inequality is $n<3$.