# How do you solve (1/3)^x-9<18 using a graph?

Jan 23, 2018

See below.

#### Explanation:

$\left(\frac{1}{3}\right) - 9 < 18 \textcolor{w h i t e}{8888}$ , $\left(\frac{1}{3}\right) - 27 < 0$

First Graph $y = \left(\frac{1}{3}\right) - 27$.

This will give you the boundary between the included and excluded regions. Remember to use a dashed line, as this is a less than and not a less than or equal to inequality, so the line will not be an included region.

With the equation plotted, you will have to possible regions. These have been marked A and B on the graph. We need to test a pair of coordinates in each region to see which region satisfies the inequality.

Region A

coordinates $\left(5 , 10\right)$

${\left(\frac{1}{3}\right)}^{5} - 27 < 10$

$243 - 27 < 10$

$\frac{1}{243} - 27 < 10 \textcolor{w h i t e}{8888}$ TRUE

A is an included region.

Now we have found our region, we do not really need to test B. Will will check this just for clarity.

Region B

coordinates $\left(5 , - 40\right)$

${\left(\frac{1}{3}\right)}^{5} - 27 < - 40$

$\frac{1}{243} - 27 < - 40 \textcolor{w h i t e}{8888}$FALSE

B is an excluded region.