Question

choose which graph correctly solves the following system: (chapter 3)
$y \leq -x - 1$
$y < -3$
a)
graph a
b)
graph b
c)
graph c
d)
graph d

Explanation:

Step1: Analyze \( y \leq -x - 1 \)

The inequality \( y \leq -x - 1 \) has a boundary line \( y = -x - 1 \) (slope -1, y - intercept -1). Since it's \( \leq \), the line is solid, and we shade below the line (because \( y \) is less than or equal to the linear expression).

Step2: Analyze \( y < -3 \)

The inequality \( y < -3 \) has a horizontal dashed line \( y = -3 \) (since it's \( < \), not \( \leq \)), and we shade below this line (where \( y \) values are less than -3).

Step3: Match with Graphs

• Check the boundary lines: \( y = -x - 1 \) (solid) and \( y = -3 \) (dashed).
• Check the shading: Below \( y = -x - 1 \) and below \( y = -3 \).
• Looking at the options, Option D has the solid line \( y = -x - 1 \), dashed line \( y = -3 \), and shading in the region that satisfies both inequalities (below both lines).

Answer:

D (The graph labeled D)