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 3 k-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 \ 3 k-3[/tex].
D. Inequality 2 is incorrect; it should be [tex]g \geq 3 k-3[/tex].
Answer (1)
Inequality 1 is incorrect.
The correct inequality should be g ≤ 80 .
Inequality 2 is correct.
The answer is: Inequality 1 is incorrect; it should be g ≤ 80 .
Explanation
Problem Analysis Let's analyze the given information to determine the correct inequalities.
Analyzing Inequality 1 Kyle states his grandmother is not more than 80 years old. This means her age, g , is less than or equal to 80. Therefore, the correct inequality should be g ≤ 80 . Jane wrote 80"> g > 80 , which is incorrect.
Analyzing Inequality 2 Kyle also says his grandmother's age is at most 3 years less than 3 times his own age. Let Kyle's age be k . Then 3 times Kyle's age is 3 k . 3 years less than that is 3 k − 3 . Grandmother's age, g , is at most this value, so g ≤ 3 k − 3 . Jane wrote g ≤ 3 k − 3 , which is correct.
Conclusion Therefore, Inequality 1 is incorrect and should be g ≤ 80 .
Examples
Understanding inequalities is crucial in various real-life scenarios. For instance, when budgeting, you might have an inequality representing that your expenses must be less than or equal to your income. Similarly, in cooking, you might need to ensure the amount of ingredients used stays within a certain range to achieve the desired taste. In this problem, we use inequalities to represent age constraints, which is a common application in demographic studies and statistical analysis.
