Chapter 8.2, Problem 27PPS

To determine

To find: the solution of the given inequality.

Answer to Problem 27PPS

The solution of the inequality is $d\ge -1$ .

Explanation of Solution

Given information:

The given inequality is ${\left(\frac{1}{27}\right)}^{2d-2}\le {81}^{d+4}$

If the base of any two exponential functions are equal then the exponent of the function are also equal. If $a^x=a^y$ then $x=y$ .

Consider the inequality.

  ${\left(\frac{1}{27}\right)}^{2d-2}\le {81}^{d+4}$ .

Solve the inequality.

  $\begin{array}{c}{\left(\frac{1}{27}\right)}^{2d-2}\le {81}^{d+4}\\ {\left(3^3\right)}^{2d-2}\le {\left(3^4\right)}^{d+4}\\ 3^{-6d+6}\le 3^{4d+16}\\ -6d+6\le 4d+16\end{array}$

Further solve the above inequality.

  $\begin{array}{c}-6d+6\le 4d+16\\ -16+6\le 4d+6d\\ -10\le 10d\\ d\ge -1\end{array}$

Therefore, the solution of the inequality is $d\ge -1$ .