Question

Solve the inequality ${32}^{5p+2}\ge {16}^{5p}$. Check your solution.

Short answer

The solution for the inequality is $p\ge -2$.

Step-by-step solution

Step 1. Write down the given inequality.

The given inequality is ${32}^{5p+2}\ge {16}^{5p}$.

Step 2. Simplify the inequality.

The given inequality ${32}^{5p+2}\ge {16}^{5p}$is solved as:

$\begin{array}{l}{32}^{5p+2}\ge {16}^{5p}\mathrm{....}\left(\text{Given}\right)\\ {\left(2^5\right)}^{5p+2}\ge {\left(2^4\right)}^{5p}\\ {\left(2\right)}^{25p+10}\ge {\left(2\right)}^{20p}\\ 25p+10\ge 20p\\ 5p\ge -10\\ p\ge -2\end{array}$

Step 3. Verify the result for obtained value.

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

$\begin{array}{l}{32}^{5p+2}\ge {16}^{5p}\mathrm{....}\left(\text{Given}\right)\\ {32}^{5\left(-2\right)+2}\ge {16}^{5\left(-2\right)}\\ {32}^{-8}\ge {16}^{-10}\\ {\left(2^5\right)}^{-8}\ge {\left(2^4\right)}^{-10}\\ {\left(2\right)}^{-40}\ge {\left(2\right)}^{-40}\mathrm{....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $p=-2$.

Step 4. Conclusion.

The solution for the given inequality is $p\ge -2$.