To solve the problem, we must first find the slope of the line containing (4, -2) and (x, -6). The slope of a line is calculated as the change in y-coordinates divided by the change in x-coordinates, or (y2 - y1) / (x2 - x1). In this case, the slope of the line containing (4, -2) and (x, -6) is (-6 - (-2)) / (x - 4) = -4 / (x - 4)
Since the line containing (4, -2) and (x,-6) is perpendicular to the line containing (-2,-9) and (3,-4), we know that the product of the slopes of these two lines is -1. We can use that information to find the slope of the second line: -1 = (-4 / (x - 4)) * m, where m is the slope of the second line. Solving for m, we get m = -1 / (-4 / (x - 4)) = (x - 4) / 4.
We can now use the point-slope form of a linear equation to find the equation of the line containing (-2,-9) and (3,-4). The point-slope form is y - y1 = m(x - x1). In this case, we can use (-2,-9) as (x1, y1) and (x - 4) / 4 as m:
y - (-9) = (x - 4) / 4 * (x + 2)
Simplifying we get y = (x-4)/4 + (-9) = (x-4)/4 - 9
Now we can substitute x = -2 and y = -9 to check if it belongs to this line:
-9 = (-2 - 4)/4 - 9
-9 = -2/4 - 9
-9 = -0.5 - 9
-9 = -9.5
It match so this line is the correct one.
The value of x that satisfies the given conditions is x = -2
You can graph this line on a separate sheet of paper by plotting the point (-2, -9) and using the slope (x-4)/4 to find additional points on the line.