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 because it states 80"> g > 80 , but the grandmother's age should be less than or equal to 80.
The correct inequality 1 is g ≤ 80 .
Inequality 2, g ≤ 3 k − 3 , is correct.
Therefore, the answer is that Inequality 1 is incorrect and should be g ≤ 80 .
Explanation
Problem Analysis Let's analyze the given information to determine the correct inequalities.
Analyzing Inequality 1 Kyle's grandmother is not more than 80 years old. This means her age, g , is less than or equal to 80. So, the correct inequality is g ≤ 80 . Jane wrote 80"> g > 80 , which is incorrect.
Analyzing Inequality 2 Kyle 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 . So, the inequality is g ≤ 3 k − 3 . Jane wrote g ≤ 3 k − 3 , which is correct.
Conclusion Therefore, Inequality 1 is incorrect, and it should be g ≤ 80 .
Examples
Understanding inequalities is crucial in various real-life scenarios. For instance, when budgeting, you might say your expenses ( e ) should be no more than your income ( i ), which translates to e ≤ i . Similarly, in cooking, you might need to ensure the oven temperature ( t ) is within a certain range, say between 150°C and 200°C, represented as 150 ≤ t ≤ 200 . These simple inequalities help manage constraints and ensure desired outcomes.
