Question

select the correct answer.

graph the following system of inequalities.

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$$\begin{cases} y \\ge \\frac{1}{3}x - 2 \\\\ y < -4x - 2 \\end{cases}$$

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Explanation:

🆕 New Concept Discovered: Graphing Systems of Inequalities
Finding where shaded regions overlap on a coordinate plane.

Step 1: Analyze the boundary lines

To graph the system, we first look at the boundary lines of each inequality:

1. For \( y \ge \frac{1}{3}x - 2 \):

• The boundary line is \( y = \frac{1}{3}x - 2 \).
• It has a \( y \)-intercept at \( (0, -2) \) and a slope of \( \frac{1}{3} \) (rise 1, run 3).
• Because the inequality symbol is \( \ge \) (greater than or equal to), the line is solid, and we shade above the line.

1. For \( y < -4x - 2 \):

• The boundary line is \( y = -4x - 2 \).
• It has a \( y \)-intercept at \( (0, -2) \) and a slope of \( -4 \) (down 4, run 1).
• Because the inequality symbol is \( < \) (less than), the line is dashed, and we shade below the line.

Step 2: Identify the intersection point

Both boundary lines share the same \( y \)-intercept:

$$ (0, -2) $$

This is the point where the solid line and the dashed line intersect.

Step 3: Determine the correct shaded region

To find the solution set of the system, we look for the region where the individual shaded areas overlap:

• Above the solid line \( y = \frac{1}{3}x - 2 \)
• Below the dashed line \( y = -4x - 2 \)

We can test a point in this region, such as \( (-2, -2) \):

• First inequality: \( -2 \ge \frac{1}{3}(-2) - 2 \implies -2 \ge -\frac{8}{3} \approx -2.67 \) (True)
• Second inequality: \( -2 < -4(-2) - 2 \implies -2 < 6 \) (True)

The correct graph will show:

1. A solid line passing through \( (0, -2) \) and \( (3, -1) \).
2. A dashed line passing through \( (0, -2) \) and \( (-1, 2) \).
3. The shaded solution region located above the solid line and below/to the left of the dashed line (containing points like \( (-2, -2) \)).

Answer:

The correct graph is the one that features:

• A solid line for \( y = \frac{1}{3}x - 2 \)
• A dashed line for \( y = -4x - 2 \)
• Both lines intersecting at \( (0, -2) \)
• Shading in the region that is above the solid line and below (to the left of) the dashed line.