Exponential Inequalities Solutions

• Exponential inequalities are solved using the    same process     as solving equations.

• Remember, if you multiply or divide by a negative number the    direction     of the inequality symbol changes.

Example 3

Solve.

 $8^{x+3}\le {\left(\frac{1}{64}\right)}^{ x–2}$


$\begin{array}{rcl}\left(\frac{1}{64}\right)& =& 8^{–2}\\ 8^{x+3}& \le & 8^{–2\left(x–2\right)}\\ x+3& \le & –2\left(x–2\right)\\ x+3& \le & –2x+4\\ 3x& \le & 1\\ x& \le & \frac{1}{3}\end{array}$

Note

You can also solve this problem using the common base, 2. Both methods of solving will result in the same value.

Example 4

Solve.  

${1000}^{x+2}>{10}^{–3x+9}$


$\begin{array}{rcl}1000& =& {10}^3\\ {10}^{3\left(x+2\right)}& >& {10}^{–3x+9}\\ 3\left(x+2\right)& >& –3x+9\\ 3x+6& >& –3x+9\\ 6x& >& 3\\ x& >& \frac{1}{2}\end{array}$