Question

\\y < -\frac{1}{2}x + 4\\
\\y \ge 3x - 3\\

Explanation:

Identify the boundary lines

We analyze the given system of inequalities:

$$
LATEXBLOCK0
$$

The first boundary line is \(y = -\frac{1}{2}x + 4\). It has a \(y\)-intercept at \((0, 4)\) and a slope of \(-\frac{1}{2}\). Since the inequality is strict (\(<\)), this boundary is represented by a dashed line.
The second boundary line is \(y = 3x - 3\). It has a \(y\)-intercept at \((0, -3)\) and a slope of \(3\). Since the inequality is non-strict (\(\ge\)), this boundary is represented by a solid line.

Determine the shaded regions

For the first inequality, \(y < -\frac{1}{2}x + 4\), we shade the region below the dashed line.
For the second inequality, \(y \ge 3x - 3\), we shade the region above and to the left of the solid line.

Find the intersection of the regions

The solution set of the system is the region where the two shaded areas overlap. This is the region that is simultaneously below the dashed line \(y = -\frac{1}{2}x + 4\) and above/left of the solid line \(y = 3x - 3\).
Looking at the graph in the image, the purple shaded region represents the area below the dashed line, and the green shaded region represents the area to the right of the solid line. The overlapping region is the dark-shaded region at the bottom-center.
To find a point in the solution set, we choose a coordinate pair \((x, y)\) that lies strictly within this overlapping region.

Select a test point

Let's test the point \((1, 1)\):

1. For \(y < -\frac{1}{2}x + 4\):

$$ 1 < -\frac{1}{2}(1) + 4 \implies 1 < 3.5 \quad (\text{True}) $$

1. For \(y \ge 3x - 3\):

$$ 1 \ge 3(1) - 3 \implies 1 \ge 0 \quad (\text{True}) $$

Since both inequalities are satisfied, the point \((1, 1)\) is in the solution set.
Another clear integer point in the overlapping region is \((0, 0)\):

1. \(0 < 4\) (True)
2. \(0 \ge -3\) (True)

Thus, \((0, 0)\) is also a valid solution.

Answer:

A point in the solution set is \((0, 0)\) (or any other point in the overlapping region, such as \((1, 1)\)).