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 (assuming the x - axis is years and we observe the curves of passion and intimacy), we need to find the range of x (years) where the passion curve is above the intimacy curve. Let's assume from the graph, the point where passion and intimacy intersect is at \(x = 8\) (or some value, but typically in such love - component graphs, the intersection is around 8 years). Before that point, passion is greater than intimacy. Also, the domain of years in a relationship is likely from \(0\) to \(10\) (or a reasonable positive range).

Step2: Set - Builder Notation

Set - builder notation for the set of years \(x\) where passion is greater than intimacy is \(\{x|0 < x < 8, x\in\mathbb{R}\}\) (assuming \(x\) is a real number representing years, and from the graph's context, the relevant interval is from \(0\) to \(8\) years approximately).

Answer:

\(\{x|0 < x < 8\}\) (The answer may vary slightly depending on the exact intersection point from the graph, but this is the general form. If the graph shows intersection at \(x = 8\), then the set is all real numbers \(x\) such that \(0\) is less than \(x\) and \(x\) is less than \(8\).)