Question

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 k, kyle’s age, and g, kyle’s grandmother’s age.
inequality 1: ( g > 80 )
inequality 2: ( g leq 3k - 3 )
which inequality did jane write incorrectly, and how could it be corrected?

• inequality 1 is incorrect; it should be ( g leq 80 ).
• inequality 1 is incorrect; it should be ( g geq 80 ).
• inequality 2 is incorrect; it should be ( g < 3k - 3 ).
• inequality 2 is incorrect; it should be ( g geq 3k - 3 ).

Explanation:

1. Analyze Inequality 1: Kyle says his grandmother is not more than 80, so \( g \leq 80 \), but Jane wrote \( g > 80 \), so Inequality 1 is wrong.
2. Analyze Inequality 2: "At most 3 years less than 3 times his age" means \( g \leq 3k - 3 \) (since "at most" is \( \leq \)), so Inequality 2 is correct.
3. Check options: The first option says Inequality 1 is incorrect and should be \( g \leq 80 \), which matches our analysis. Other options have errors (e.g., second option says \( g \geq 80 \) which contradicts "not more than 80"; third and fourth options incorrectly claim Inequality 2 is wrong).

Answer:

A. Inequality 1 is incorrect; it should be \( g \leq 80 \).