Question

which system of linear inequalities is represented by the graph?

• ( y ge x - 2 ) and ( y le x + 1 )
• ( y < x - 2 ) and ( y > x + 1 )
• ( y le x - 2 ) and ( y ge x + 1 )
• ( y > x - 2 ) and ( y < x + 1 )

Explanation:

Step1: Identify boundary lines

The two lines are $y = x + 1$ and $y = x - 2$, both are solid lines (so inequalities use $\geq$ or $\leq$).

Step2: Test shading for $y=x+1$

Pick a point in the shaded region above $y=x+1$, e.g., $(0,2)$: $2 \geq 0+1$, so $y \geq x+1$.

Step3: Test shading for $y=x-2$

Pick a point in the shaded region below $y=x-2$, e.g., $(0,-3)$: $-3 \leq 0-2$, so $y \leq x-2$.

Step4: Match to options

Combine the inequalities: $y \leq x-2$ and $y \geq x+1$.

Answer:

$y \leq x-2$ and $y \geq x+1$ (third option)