Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
$y < -\frac{1}{2}x + 4$
$y \leq x + 7$

Explanation:

Step1: Identify boundary lines

For $y < -\frac{1}{2}x + 4$: boundary is dashed line $y = -\frac{1}{2}x + 4$ (slope $-\frac{1}{2}$, y-intercept 4).
For $y \leq x + 7$: boundary is solid line $y = x + 7$ (slope 1, y-intercept 7).

Step2: Shade solution regions

Shade below dashed line $y = -\frac{1}{2}x + 4$ (for $y < -\frac{1}{2}x + 4$).
Shade below solid line $y = x + 7$ (for $y \leq x + 7$).

Step3: Find overlapping region

The solution set is the intersection of the two shaded areas (the lighter blue region in the given graph).

Step4: Pick a point in overlap

Choose any point within the overlapping shaded area.

Answer:

A point in the solution set is $(0, 0)$ (other valid points include $(2, 1)$, $(-1, 0)$, etc.)