Question - Equation of a Line and Inequalities in Shaded Regions
Solution:
I'll help you to determine the equation of the line shown in part a of the image, and then we can use this information to write inequalities for the shaded regions shown in parts b and c.To find the equation of a line, we need to identify two key components: the slope and the y-intercept. In the coordinate grid provided, the line crosses the y-axis at (0, -2), which means the y-intercept is -2. The line also looks as if it passes through another point with integer coordinates, which we can use to determine the slope.I see that the line goes through (2, 2) along with (0, -2). The slope ($$m$$) is the change in y over the change in x when moving from one point to another:\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-2)}{2 - 0} \]\[ m = \frac{4}{2} = 2 \]So the slope of the line is 2. Now that we have both the slope and the y-intercept, we can write the equation of the line in slope-intercept form (y = mx + b):\[ y = 2x - 2 \]For part b, if the graph is shaded above this line, then the inequality representing the shaded region would use a "greater than" symbol, because the y-values are greater than the y-values on the line for any given x-value.The inequality for the graph in part b would be:\[ y > 2x - 2 \]For part c, if the graph is shaded below this line, then the inequality representing the shaded region would use a "less than" symbol, because the y-values are less than the y-values on the line for any given x-value.The inequality for the graph in part c would be:\[ y < 2x - 2 \]
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