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 intimacy is greater than commitment. the years for which intimacy is greater than commitment in a relationship is {x\\(\square\\)}.

Explanation:

Step1: Analyze the Graph

We look at the graph of intimacy (blue curve) and commitment (black curve). We need to find the \( x \)-values (years) where the intimacy curve is above the commitment curve. From the graph, we can see that the two curves intersect at \( x = 4 \) (approximately, by looking at the graph where the blue and black curves meet) and we need to find the interval where intimacy > commitment. Wait, actually, looking at the graph, the intimacy (blue) and commitment (black) - let's check the intersection point. Wait, maybe I misread. Wait, the x - axis is years, y - axis is level of intensity. Let's see: the intimacy curve and commitment curve - when does intimacy > commitment? Let's see the intersection point. From the graph, the two curves (intimacy and commitment) intersect at \( x = 4 \)? Wait, no, maybe at \( x = 4 \)? Wait, actually, looking at the graph, the intimacy (blue) and commitment (black) - let's see the x - values. Wait, the commitment curve (black) and intimacy (blue) - when x is less than 4? Wait, no, maybe I got it wrong. Wait, let's re - examine. The problem is to find when intimacy (blue) is greater than commitment (black). So we need to find the x - values (years) where the blue curve (intimacy) is above the black curve (commitment). From the graph, we can see that the two curves intersect at \( x = 4 \) (approximately, by looking at the grid). Wait, no, maybe the intersection is at \( x = 4 \)? Wait, actually, looking at the graph, the intimacy (blue) and commitment (black) - when x is less than 4? Wait, no, maybe I made a mistake. Wait, let's think again. The x - axis is years (0 to 10), y - axis is intensity. The intimacy curve (blue) and commitment curve (black) - let's see the points. At x = 4, maybe they cross? Wait, no, looking at the graph, the commitment curve (black) starts to rise, and intimacy (blue) also rises. Wait, maybe the intersection is at x = 4? Wait, no, perhaps the correct intersection is at x = 4, and for x < 4, intimacy is greater than commitment? Wait, no, maybe I have it reversed. Wait, let's check the y - values. Wait, the blue curve is intimacy, black is commitment. Let's take x = 2: intimacy (blue) is at some value, commitment (black) is lower. At x = 4: maybe they are equal? Then, after x = 4, commitment is higher? Wait, no, maybe the other way. Wait, the problem says "intimacy is greater than commitment". So we need to find the set of x (years) where \( \text{intimacy}(x)>\text{commitment}(x) \). From the graph, we can see that the two curves (intimacy and commitment) intersect at \( x = 4 \), and for \( x < 4 \), intimacy is above commitment? Wait, no, maybe I got the intersection wrong. Wait, maybe the intersection is at x = 4, and for x < 4, intimacy > commitment? Wait, no, let's look at the graph again. The x - axis is years (0,2,4,6,8,10). The intimacy curve (blue) and commitment curve (black) - at x = 4, maybe they cross. So the solution is the set of x such that \( x < 4 \)? Wait, no, maybe I made a mistake. Wait, perhaps the correct intersection is at x = 4, and for x < 4, intimacy is greater than commitment. Wait, but let's confirm. So the set - builder notation is \( \{x|x < 4\} \)? Wait, no, maybe the intersection is at x = 4, and we need to find the interval where intimacy > commitment. Wait, maybe the graph shows that the two curves intersect at x = 4, and for x < 4, intimacy is above commitment. So the inequality is \( x < 4 \), so the set - builder notation is \( \{x|x\in\mathbb{R},x < 4\} \), but since x represents years in a relationship, x is non - negative (x ≥ 0). Wait, but the problem doesn't specify the domain, but from the graph, x is from 0 to 10. But the main thing is to find where intimacy > commitment. So from the graph, the two curves (intimacy and commitment) intersect at x = 4, so for x < 4, intimacy is greater than commitment.

Step2: Write the Set - Builder Notation

Set - builder notation has the form \( \{x|P(x)\} \), where \( P(x) \) is the predicate. Here, the predicate is "intimacy is greater than commitment", which translates to \( x < 4 \) (assuming the intersection is at x = 4). So the set - builder notation is \( \{x|x < 4\} \). Wait, but we need to consider the domain of x (years in a relationship, so x ≥ 0, but maybe the problem just wants the inequality based on the graph intersection). So the inequality is \( x < 4 \), so the set - builder notation is \( \{x|x < 4\} \).

Answer:

\( x < 4 \) (So the set - builder notation is \( \{x|x < 4\} \))