Question

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

Explanation:

Step1: Identify boundary line

The boundary is $y = -\frac{1}{2}x - 1$, a solid line (due to $\leq$).

Step2: Find intercepts for plotting

• y-intercept: Set $x=0$, $y = -\frac{1}{2}(0)-1 = -1$. Point: $(0, -1)$
• x-intercept: Set $y=0$, $0 = -\frac{1}{2}x -1 \implies x = -2$. Point: $(-2, 0)$

Step3: Plot and draw the line

Plot $(0,-1)$ and $(-2,0)$, draw a solid straight line through them.

Step4: Determine shading region

Test $(0,0)$: $0 \leq -\frac{1}{2}(0)-1 \implies 0 \leq -1$, which is false. Shade the region below the line (opposite of the test point).

Answer:

Graph of $y \leq -\frac{1}{2}x - 1$ (solid line with shading below)