不等式y大于3x加2表示一个线性不等式。边界线y等于3x加2的斜率为3，y截距为2。由于不等式使用的是大于号，而不是大于等于，所以边界线用虚线表示，表明它不包含在解集中。解的区域是这条虚线上方的区域，我们用阴影来表示所有满足y大于3x加2的点。
To determine which region satisfies the inequality, we use the test point method. First, we test the point (0, 0). Substituting into the inequality: 0 is greater than 3 times 0 plus 2, which is 0 greater than 2. This is false, so point (0, 0) is not in the solution region. Next, we test point (0, 5). Substituting: 5 is greater than 3 times 0 plus 2, which is 5 greater than 2. This is true, so point (0, 5) is in the solution region. This confirms that the region above the dashed line is the correct solution.
It's important to understand the difference between strict and non-strict inequalities. The inequality y greater than 3x plus 2 uses a dashed line because points on the boundary are not included in the solution. In contrast, y greater than or equal to 3x plus 2 would use a solid line, indicating that points on the boundary line are included. For example, the point (0, 2) lies exactly on the line y equals 3x plus 2. This point satisfies y greater than or equal to 3x plus 2, but does not satisfy y greater than 3x plus 2.
Linear inequalities have numerous real-world applications across various fields. In business, they can represent profit constraints or production limitations. In engineering, they define safety margins and operating ranges. For example, consider a scenario where x represents hours worked and y represents total earnings in dollars. The inequality y greater than 3x plus 2 means earning more than 3 dollars per hour plus a 2 dollar bonus. The shaded region shows all combinations of hours and earnings that satisfy this condition. A point like 4 hours and 18 dollars falls within the feasible region, indicating this earning scenario exceeds the minimum requirement.
To summarize, the graph of y greater than 3x plus 2 represents a linear inequality with several key characteristics. The boundary line y equals 3x plus 2 has a slope of 3 and y-intercept of 2. Since we use the strict inequality symbol greater than, the boundary line is dashed, indicating points on the line are not included in the solution set. The solution region is the area above this dashed line, which we can verify using test points. This type of inequality appears frequently in real-world applications across mathematics, science, and engineering. Remember that the key difference between greater than and greater than or equal to is whether the boundary line is dashed or solid.