Equation of a Line and Inequalities in Shaded Regions
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:
\[ y < 2x - 2 \]
