Chapter 8.2, Problem 26PPS

To determine

To find: the solution of the given inequality.

Answer to Problem 26PPS

The solution of the inequality is $c>\frac{3}{4}$ .

Explanation of Solution

Given information:

The given inequality is ${\left(\frac{1}{64}\right)}^{c-2}>{32}^{2c}$

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.

  ${\left(\frac{1}{64}\right)}^{c-2}>{32}^{2c}$ .

Solve the inequality.

  $\begin{array}{c}{\left(\frac{1}{64}\right)}^{c-2}>{32}^{2c}\\ {\left(2^6\right)}^{-c+2}>{\left(2^5\right)}^{2c}\\ 2^{-6c+12}>2^{10c}\\ -6c+12>10c\end{array}$

Further solve the above inequality.

  $\begin{array}{c}-6c+12>10c\\ 12>10c+6c\\ c>\frac{12}{16}\\ >\frac{3}{4}\end{array}$

Therefore, the solution of the inequality is $c>\frac{3}{4}$ .