Question

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

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which graph represents the system of inequalities?

w.
x.
y.
z.

Explanation:

🆕 New Concept Discovered: Graphing Systems of Linear Inequalities
Shading overlapping regions of boundary lines

Step 1: Identify the inequalities

The system of inequalities shown at the top of the image is:

$$
LATEXBLOCK0
$$

Step 2: Analyze the boundary lines and shading

For the first inequality, \( y \ge \frac{1}{3}x - 2 \):

• The boundary line is solid because of the "\(\ge\)" symbol.
• The line has a \(y\)-intercept at \((0, -2)\) and a positive slope of \(\frac{1}{3}\).
• The shading is above this solid line because of the "\(\ge\)" symbol.

For the second inequality, \( y < -4x - 2 \):

• The boundary line is dashed because of the "\(<\)" symbol.
• The line has a \(y\)-intercept at \((0, -2)\) and a steep negative slope of \(-4\).
• The shading is below (to the left of) this dashed line because of the "\(<\)" symbol.

Step 3: Find the overlapping region

We need to find the graph where the shaded region is simultaneously above the solid line \( y = \frac{1}{3}x - 2 \) and below/left of the dashed line \( y = -4x - 2 \):

• Graph W: The shaded region is above the solid line and to the right of the dashed line.
• Graph X: The shaded region is above the solid line and to the left of the dashed line. This matches both of our conditions.
• Graph Y: The shaded region is below the solid line.
• Graph Z: The shaded region is below the solid line.

Therefore, Graph X represents the correct solution set.

Answer:

X