Question

solve the following system of inequalities graphically on the set of axes below, state the coordinates of a point in the solution set.
$y > -2x - 2$
$y \geq x - 8$

Explanation:

Step1: Graph first inequality

First, graph the line $y=-2x-2$ as a dashed line (since the inequality is $>$). Test a point like $(0,0)$: $0 > -2(0)-2$ simplifies to $0 > -2$, which is true, so shade the region above this line.

Step2: Graph second inequality

Next, graph the line $y=x-8$ as a solid line (since the inequality is $\geq$). Test a point like $(0,0)$: $0 \geq 0-8$ simplifies to $0 \geq -8$, which is true, so shade the region above this line.

Step3: Identify overlapping region

The solution set is the overlapping shaded region of the two inequalities.

Step4: Pick a point in the overlap

Choose a point from the overlapping region, e.g., $(0,0)$. Verify:
For $y > -2x-2$: $0 > -2(0)-2 \implies 0 > -2$ (true)
For $y \geq x-8$: $0 \geq 0-8 \implies 0 \geq -8$ (true)

Answer:

$(0, 0)$
(Note: Any point in the overlapping shaded region is a valid answer, e.g., $(1,1)$, $(2,0)$ are also acceptable.)