Question

find the graph of this inequality
y ≤ 3x + 1

Explanation:

Step1: Analyze the boundary line

The inequality is \( y \leq 3x + 1 \). First, consider the boundary line \( y = 3x + 1 \). The slope \( m = 3 \) (positive, so the line rises from left to right) and the y - intercept \( b = 1 \) (the line crosses the y - axis at \( (0,1) \)). Since the inequality is \( \leq \), the boundary line should be solid (to include the points on the line).

Step2: Determine the shaded region

To find which side of the line to shade, we can use a test point. A common test point is \( (0,0) \) (as long as it's not on the line). Substitute \( x = 0 \) and \( y = 0 \) into the inequality: \( 0\leq3(0)+1\), which simplifies to \( 0\leq1 \). This is a true statement. So, we shade the region that includes the point \( (0,0) \).

For a line with a positive slope (\( m = 3 \)), the region that includes \( (0,0) \) (which is below the line \( y = 3x + 1 \) when \( x = 0 \), since \( 0<1 \)) will be the area below the solid line with a positive slope.

(Note: Since the graphs are not fully visible with coordinates, but based on the slope and the test point, we can infer the correct graph. The line \( y = 3x+1 \) has a positive slope, solid line, and shaded below the line (including the line) as the test point \( (0,0) \) satisfies the inequality.)

Answer:

The graph should be a solid line with a positive slope (slope of 3) crossing the y - axis at (0,1), and the region below (and including) the line is shaded. (To match the given options, we would identify the graph with a solid line \( y = 3x + 1 \) and shading below the line. If we assume the third graph (right - most) has the correct line and shading, but since the graphs are partially visible, the key is the solid line with positive slope and shading below.)