Question

Solve the inequality $2^{2n}\le \frac{1}{16}$. Check your solution.

Short answer

The solution for the inequality is $n\le -2$.

Step-by-step solution

Step 1. Write down the given inequality.

The given inequality is $2^{2n}\le \frac{1}{16}$.

Step 2. Simplify the inequality.

The given inequality $2^{2n}\le \frac{1}{16}$is solved as:

$\begin{array}{l}{2}^{2n}\le \frac{1}{16}\mathrm{....}\left(\text{Given}\right)\\ 2^{2n}\le {16}^{-1}\\ 2^{2n}\le {\left(2^4\right)}^{-1}\\ 2^{2n}\le 2^{-4}\\ 2n\le -4\\ n\le -2\end{array}$

Step 3. Verify the result for obtained value.

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

$\begin{array}{l}{2}^{2n}\le \frac{1}{16}\mathrm{....}\left(\text{Given}\right)\\ 2^{2\left(-2\right)}\le \frac{1}{16}\\ 2^{-4}\le \frac{1}{16}\\ \frac{1}{2^4}\le \frac{1}{16}\\ \frac{1}{16}\le \frac{1}{16}\mathrm{....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $n\le -2$.

Step 4. Conclusion.

The solution for the given inequality is $n\le -2$.