Answer :

GinnyAnswer
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.

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