Question

1. a system of inequalities is given and graphed below. which shaded region of the graph is the solution to this system?

$y < x - 1$
$-x + 2y \geq 1$

Explanation:

Step1: Rearrange first inequality

Rewrite $y < x - 1$ as $y - x < -1$.

Step2: Rearrange second inequality

Rewrite $-x + 2y \leq 1$ as $2y - x \leq 1$.

Step3: Test origin (0,0)

For $y < x - 1$: $0 < 0 - 1 \implies 0 < -1$, false. So solution is opposite side of line from origin.
For $-x + 2y \leq 1$: $0 + 0 \leq 1 \implies 0 \leq 1$, true. So solution is same side of line as origin.

Step4: Identify overlapping region

The overlap of the two solution regions is Region II.

Answer:

Region II