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)
The problem states Kyle's grandmother is not more than 80 years old, which translates to g ≤ 80 .
Jane incorrectly wrote Inequality 1 as 80"> g > 80 .
Inequality 2, g ≤ 3 k − 3 , is correctly written.
Therefore, Inequality 1 is incorrect and should be g ≤ 80 .
Explanation
Problem Analysis Let's analyze the given information to determine which inequality Jane wrote incorrectly.
Analyzing Inequality 1 Kyle states that his grandmother is 'not more than 80 years old'. This means the grandmother's age, represented by g , must be less than or equal to 80. Therefore, the correct inequality should be: g ≤ 80 Jane wrote Inequality 1 as 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'. This means g is less than or equal to 3 k − 3 . Therefore, the inequality is: g ≤ 3 k − 3 Jane wrote Inequality 2 as g ≤ 3 k − 3 , which is correct.
Conclusion Therefore, Inequality 1 is incorrect, and it should be corrected to g ≤ 80 .
Examples
Understanding inequalities is crucial in various real-life scenarios. For instance, when budgeting, you might set an inequality to ensure your expenses don't exceed your income. Similarly, in cooking, you might use inequalities to maintain the correct ratios of ingredients. In this case, the problem demonstrates how to translate verbal statements into mathematical inequalities, a skill applicable in finance, science, and everyday decision-making.
