The graph of the linear inequality y < 7x + 4 represents a region in the two-dimensional Cartesian coordinate system. To understand this graph, we start by analyzing the corresponding linear equation, which is y = 7x + 4.
1. **Graphing the Boundary Line:**
First, we plot the boundary line of the equation y = 7x + 4. This line has a slope of 7 and a y-intercept of 4. This means that for every unit increase in x, y increases by 7 units. The line itself is solid if the inequality were
“y ≤ 7x + 4” or “y ≥ 7x + 4”, but since we have a strict inequality (y <), we will use a dashed line to indicate that points on the line are not included in the solution set.
2. **Choosing the Correct Region:**
To determine which side of the line represents the solution to the inequality, we can use a test point not on the line, such as (0,0). Plugging this point into the inequality:
0 < 7(0) + 4
0 < 4 This inequality is true, which means the region containing the point (0,0) satisfies the inequality. Thus, we shade the area below the dashed line, where y is less than the linear expression.
3. **Final Visualization:**
Overall, the graph of <code>y < 7x + 4</code> consists of a dashed line representing the boundary y = 7x + 4 and a shaded region beneath this line, extending infinitely in all directions but never including the points on the line itself. Any point (x, y) within this shaded area will satisfy the inequality.
In summary, the correct description of the graph of the inequality y < 7x + 4 includes a dashed boundary line and a shaded region below this line.