Question - Determining Equations of Lines and Inequalities
Solution:
To find the equation of the line in part a, we need to determine the slope and the y-intercept.The equation of a line in slope-intercept form is:y = mx + bwhere m is the slope of the line, and b is the y-intercept.We can find the slope by looking at two points on the line and using the slope formula:slope (m) = (change in y) / (change in x)From the image, we can pick two points that the line passes through. Let's choose (0, -2) and (2, -1) since they're clearly on the grid intersections.Now, we find the slope:m = (y2 - y1) / (x2 - x1)m = (-1 - (-2)) / (2 - 0)m = (1) / (2)m = 1/2Next, we find the y-intercept (b). This is where the line crosses the y-axis. Looking at the graph, we can see that this occurs at (0, -2), so b = -2.Now we have the slope m = 1/2 and y-intercept b = -2, the equation of the line is:y = (1/2)x - 2To find the inequalities for parts b and c, we would normally look for instructions that designate whether the area above or below the line should be shaded. Since the image only shows the line and does not specify the inequalities or shaded regions, we can't determine what the inequalities would be. However, if you are being asked for the inequality that includes points below the line, the inequality would be:y ≤ (1/2)x - 2If the inequality should include points above the line, it would be:y ≥ (1/2)x - 2Remember, without additional information or context regarding which side of the line should be considered for the inequality, you cannot definitively state the inequality.
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