Kyle asks his friend Jane to guess his age and his grandmother's age. Kyle says his grandmother is not more than 80 years old. He says his grandmother's age is, at most, 3 years less than 3 times his own age.

Jane writes this system of inequalities to represent [tex]$ k $[/tex], Kyle's age, and [tex]$ g $[/tex], Kyle's grandmother's age.

Inequality 1: [tex]$ g \ \textgreater \ 80 $[/tex]
Inequality 2: [tex]$ g \leq 3k - 3 $[/tex]

Which inequality did Jane write incorrectly, and how could it be corrected?

A. Inequality 1 is incorrect; it should be [tex]$ g \leq 80 $[/tex].
B. Inequality 1 is incorrect; it should be [tex]$ g \geq 80 $[/tex].
C. Inequality 2 is incorrect; it should be [tex]$ g \ \textless \ 3k - 3 $[/tex].
D. Inequality 2 is incorrect; it should be [tex]$ g \geq 3k - 3 $[/tex].

Alright, let's break down the problem step-by-step to identify which inequality is incorrect and how it should be corrected.

Kyle gives two pieces of information about the ages:

1. His grandmother is not more than 80 years old.
2. His grandmother's age is at most 3 years less than 3 times Kyle's age.

Firstly, let's analyze the system of inequalities Jane wrote:

- Inequality 1: [tex]$ g > 80 $[/tex]
- Inequality 2: [tex]$ g \leq 3k - 3 $[/tex]

Given the information from Kyle, let's check the correctness of each inequality:

### Step 1: Analyze Inequality 1

Kyle's statement: His grandmother is not more than 80 years old.

This means the grandmother's age [tex]$ g $[/tex] should be less than or equal to 80 years. Mathematically, this is represented as:
[tex]\[ g \leq 80 \][/tex]

However, Jane wrote:
[tex]\[ g > 80 \][/tex]

This contradicts Kyle's information. Therefore, Inequality 1 is incorrect. The correct version should be:
[tex]\[ g \leq 80 \][/tex]

### Step 2: Analyze Inequality 2

Kyle's statement: His grandmother's age is at most 3 years less than 3 times Kyle's age.

This means the grandmother's age [tex]$ g $[/tex] should be less than or equal to [tex]$ 3k - 3 $[/tex] (where [tex]$ k $[/tex] is Kyle's age). Mathematically, this is represented as:
[tex]\[ g \leq 3k - 3 \][/tex]

Jane wrote:
[tex]\[ g \leq 3k - 3 \][/tex]

This matches Kyle's information. Therefore, Inequality 2 is correctly written.

### Conclusion

From the analysis:

- Inequality 1 is incorrect. The correct form should be [tex]$ g \leq 80 $[/tex].

So, the incorrect inequality is Inequality 1, and the correction for it is:
[tex]\[ g \leq 80 \][/tex]

Therefore, the correct answer is:
"Inequality 1 is incorrect; it should be [tex]$ g \leq 80 $[/tex]."