Answer :
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]."
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]."