Question

graph the inequality on the axes below.
$y \leq -\frac{1}{2}x - 1$

Explanation:

Step1: Identify boundary line form

The inequality $y \leq -\frac{1}{2}x - 1$ has a boundary line $y = -\frac{1}{2}x - 1$, which is in slope-intercept form $y=mx+b$, where $m=-\frac{1}{2}$ (slope) and $b=-1$ (y-intercept).

Step2: Plot y-intercept

The y-intercept is $(0, -1)$. Mark this point on the axes.

Step3: Use slope to find second point

Slope $m=-\frac{1}{2}=\frac{\Delta y}{\Delta x}$. From $(0, -1)$, move down 1 unit and right 2 units to get the point $(2, -2)$. Mark this point.

Step4: Draw boundary line

Since the inequality is $\leq$, draw a solid straight line through $(0, -1)$ and $(2, -2)$ (solid line includes points on the line).

Step5: Shade the solution region

Test the origin $(0,0)$: $0 \leq -\frac{1}{2}(0) -1$ simplifies to $0 \leq -1$, which is false. Shade the half-plane that does NOT contain the origin (the region below the boundary line).

Answer:

1. Solid line through points $(0, -1)$ and $(2, -2)$
2. Shaded region below the solid line