Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.

Steps to Solve

1. Rearranging the inequalities

First, note the two inequalities that we need to graph:

1. ( y > \frac{1}{6}x + 4 )

2. ( y < -\frac{4}{3}x - 5 )

3. Graphing the first inequality

To graph the first inequality, start by converting it to the equation of a line:

$$ y = \frac{1}{6}x + 4 $$

Plot this line using points, like when ( x = 0 ), ( y = 4 ) (point (0, 4)) and ( x = 6 ), ( y = 5 ) (point (6, 5)). Draw a dashed line because the inequality is strict (greater than).

3. Shading the region for the first inequality

Since the inequality is ( y > \frac{1}{6}x + 4 ), shade above the dashed line to represent all points where ( y ) is greater than the line.

4. Graphing the second inequality

Next, graph the second inequality by converting it to the equation of a line:

$$ y = -\frac{4}{3}x - 5 $$

Plot this line using points, such as when ( x = 0 ), ( y = -5 ) (point (0, -5)) and ( x = 3 ), ( y = -9 ) (point (3, -9)). Again, use a dashed line since it's a strict inequality (less than).

5. Shading the region for the second inequality

For the second inequality, ( y < -\frac{4}{3}x - 5 ), shade below the dashed line to indicate all points where ( y ) is less than the line.

6. Identifying the solution region

The solution to the system of inequalities is where the shaded regions of both inequalities intersect. This is the feasible region.

7. Choosing a point within the solution set

Select a point that lies in the intersection of the shaded regions. A suitable point could be ( (0, -6) ).