Question

question 5
graph the system of inequalities.
x - y ≤ 8
y < -\frac{1}{2}x + 3
note: after you make your two linear inequalities, use the red and blue buttons in the bottom right of the graph to ensure you correctly have a solid or dashed line. after you are satisfied with the lines, place a green dot representing the correct shaded region on the graph by clicking the graph.

Explanation:

Step1: Rewrite first inequality to slope-intercept

Rearrange $x - y \leq 8$ to solve for $y$:
$ -y \leq -x + 8 $
Multiply by -1 (reverse inequality):
$ y \geq x - 8 $
This is a solid line (due to $\geq$) with slope $1$ and y-intercept $-8$.

Step2: Rewrite second inequality (verify format)

The given inequality is $y < -\frac{1}{4}x + 3$.
This is a dashed line (due to $<$) with slope $-\frac{1}{4}$ and y-intercept $3$.

Step3: Identify shaded region

For $y \geq x - 8$: shade above the solid line.
For $y < -\frac{1}{4}x + 3$: shade below the dashed line.
The overlapping region is the solution set. A sample point in this region is $(0,0)$ (check: $0-0 \leq 8$ is true; $0 < 0 + 3$ is true).

Answer:

1. Draw a solid line for $y = x - 8$, shade the area above this line.
2. Draw a dashed line for $y = -\frac{1}{4}x + 3$, shade the area below this line.
3. Place a green dot in the overlapping shaded region (e.g., at the point $(0,0)$).