Question - Finding Equation of a Line and Inequalities
Solution:
To find the equation of the line, you need to find its slope and y-intercept.From the graph, you can see the line crosses the y-axis at the point (0, 1), which means the y-intercept is 1.To find the slope, you can pick two points on the line. The line passes through the points (0, 1) and (-2, -1) which can be clearly seen on the graph.The slope (m) is calculated as the rise over the run:\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 1}{-2 - 0} = \frac{-2}{-2} = 1. \]So, the slope is 1. With a slope of 1 and a y-intercept of 1, the equation of the line in slope-intercept form (y = mx + b) is:\[ y = 1x + 1 \]or simply:\[ y = x + 1. \]For inequalities:a) Since the line is solid, this means that the inequality includes equality; that is, points on the line satisfy the inequality. Since the area below the line is shaded, the inequality represents all points where y is less than or equal to x + 1.Therefore, the inequality for part a) is:\[ y \leq x + 1. \]b) and c) are not provided explicitly in the question, but typically they would be related to different shading regions if provided. If the shaded region was above the line for part b), for example, the inequality would be y ≥ x + 1. If the shaded region for part c) omitted points on the line, the inequality would use strict inequality signs (< or >) rather than ≤ or ≥.
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