Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
$y \geq \frac{3}{2}x - 8$
$y < -x - 3$

Explanation:

Step1: Identify line types

For $y \geq \frac{3}{2}x - 8$: solid line (≥ includes equality).
For $y < -x - 3$: dashed line (< excludes equality).

Step2: Graph boundary lines

1. For $y = \frac{3}{2}x - 8$:

Y-intercept: $(0, -8)$, slope $m=\frac{3}{2}$.

1. For $y = -x - 3$:

Y-intercept: $(0, -3)$, slope $m=-1$.

Step3: Shade solution regions

1. For $y \geq \frac{3}{2}x - 8$: shade above the solid line.
2. For $y < -x - 3$: shade below the dashed line.

Step4: Find overlapping region

The solution set is the intersection of the two shaded areas.

Step5: Pick a point in overlap

Choose a point lying in both shaded regions, e.g., $(0, -5)$.
Verify:
$ -5 \geq \frac{3}{2}(0) -8 \implies -5 \geq -8$ (true)
$ -5 < -(0) -3 \implies -5 < -3$ (true)

Answer:

A point in the solution set is $(0, -5)$ (any point in the overlapping shaded region is valid)