Question

y < -x - 1

Explanation:

Identify the boundary lines from the system

The system of inequalities is partially visible at the top:

$$
LATEXBLOCK0
$$

The first boundary line is \(y = -x - 1\), which has a \(y\)-intercept of \((0, -1)\) and a slope of \(-1\). It is represented by a dashed line because of the strict inequality \(<\).
The second boundary line is \(y = 3x + 1\), which has a \(y\)-intercept of \((0, 1)\) and a slope of \(3\). It is represented by a solid line because of the non-strict inequality (or dashed depending on the standard, but here the red/blue line is solid).

Determine the shaded region

For \(y < -x - 1\), the shaded region is below the dashed line \(y = -x - 1\).
For \(y > 3x + 1\), the shaded region is above/to the left of the line \(y = 3x + 1\).
Testing the point \((-2, -2)\):

$$
LATEXBLOCK1
$$

Thus, the region containing \((-2, -2)\) must be shaded.

Match with the given options

• Graph W: The shaded region is below the dashed line \(y = -x - 1\) and to the left of the solid line \(y = 3x + 1\). The point \((-2, -2)\) lies in this shaded region.
• Graph X: The shaded region is below both lines.
• Graph Y: The shaded region is below both lines.
• Graph Z: The shaded region is to the right of the solid line.

Therefore, Graph W represents the correct solution.

Answer:

• W. (Correct answer)
• X.
• Y.
• Z.