Question

Solve the inequality $5^{2x+3}\le 125$. Check your solution.

Short answer

The solution for the given inequality is$x\le 0$.

Step-by-step solution

Step 1. Write down the given expression.

The given inequality is $5^{2x+3}\le 125$.

Step 2. Simplify the equation.

The given inequality $5^{2x+3}\le 125$is solved as:

$\begin{array}{l}{5}^{2x+3}\le \mathrm{125....}\left(\text{Given}\right)\\ {\left(5\right)}^{2x+3}\le {\left(5\right)}^3\\ 2x+3\le 3\\ 2x\le 0\\ x\le 0\end{array}$

Step 3. Verify the result for obtained value.

Plugging $x=0$in given inequality and verify the result.

$\begin{array}{l}{5}^{2x+3}\le \mathrm{125....}\left(\text{Given}\right)\\ 5^{2\left(0\right)+3}\le \mathrm{125....}\left(\text{Plugging}x=0\right)\\ 5^3\le 125\\ 125=\mathrm{125....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $x=0$.

Step 4. Conclusion.

The solution for the given inequality is $x\le 0$.