To graph the solution of the inequality:
[tex]\[
\frac{4}{9} x - 10 > \frac{x}{3} - 12
\][/tex]
Let's solve this step-by-step:
1. Rewrite the inequality:
[tex]\[
\frac{4}{9} x - 10 > \frac{x}{3} - 12
\][/tex]
2. First, let's isolate the variable \( x \) on one side of the inequality:
Multiply both sides by 9 to eliminate the fractions:
[tex]\[
9 \left(\frac{4}{9} x - 10\right) > 9 \left(\frac{x}{3} - 12\right)
\][/tex]
Simplifying, we get:
[tex]\[
4x - 90 > 3x - 108
\][/tex]
3. Subtract \( 3x \) from both sides to combine the \( x \) terms:
[tex]\[
4x - 3x - 90 > -108
\][/tex]
Simplifying, we get:
[tex]\[
x - 90 > -108
\][/tex]
4. Add 90 to both sides to isolate \( x \):
[tex]\[
x - 90 + 90 > -108 + 90
\][/tex]
Simplifying, we get:
[tex]\[
x > -18
\][/tex]
The solution to the inequality is:
[tex]\[
x > -18
\][/tex]
### Graphing the Solution:
To graph this solution on a number line, follow these steps:
1. Draw a number line and mark the point \(-18\) on it.
2. Since \(x > -18\), this means \(x\) can be any number greater than \(-18\). Thus, place an open circle at \(-18\) to indicate that \(-18\) is not included in the solution set.
3. Shade the number line to the right of \(-18\) to indicate that all numbers greater than \(-18\) are part of the solution.
Here is what the number line would look like:
```
<---|---|---|---|---|---|---|---|---|---|---|---|--->
-21 -20 -19 (-18) -17 -16 -15 -14 -13 -12 -11 -10
o====================>
```
In this representation:
- The open circle at \(-18\) indicates that \(-18\) is not included.
- The shading to the right indicates all values greater than [tex]\(-18\)[/tex] form the solution set.
