Question

the graph to the right shows that the three components of love, namely passion, intimacy, and commitment, progress differently over time. passion peaks early in a relationship and then declines. by contrast, intimacy and commitment build gradually. use set - builder notation to write an inequality that expresses for which years in a relationship passion is greater than intimacy. the years for which passion is greater than intimacy in a relationship is {x|□}.

Explanation:

Step1: Analyze the Graph

From the graph, we observe the relationship between the level of passion and intimacy over years (\(x\)). We need to find the range of \(x\) (years) where the level of passion is greater than intimacy. Looking at the graph, the point where passion and intimacy intersect (or where passion starts to be less than intimacy) seems to be around \(x = 7\) (visually inspecting the graph, the curves of passion and intimacy cross at \(x = 7\) approximately). Before that, passion is greater than intimacy, and since the relationship starts at \(x = 0\) (years in a relationship start from 0), we consider the domain of \(x\) as non - negative (as years can't be negative in a relationship context).

Step2: Formulate the Inequality in Set - Builder Notation

Set - builder notation is of the form \(\{x| \text{condition on } x\}\). The condition here is that the number of years \(x\) is greater than or equal to 0 (since a relationship can't have negative years) and less than 7 (because at \(x = 7\) (approximately) passion and intimacy are equal, and before that passion is greater). So the set - builder notation for the years when passion is greater than intimacy is \(\{x|0\leq x < 7\}\) (assuming the intersection point is at \(x = 7\) from the graph's visual representation).

Answer:

\(\{x|0\leq x < 7\}\)