Question

1. which is the correct graph of the inequality $2x + y \geq 2$?

Explanation:

Step1: Rewrite to slope-intercept form

Start with the given inequality and solve for $y$:
$$2x + y \geq 2 \implies y \geq -2x + 2$$

Step2: Identify boundary line

The boundary line is $y = -2x + 2$, which has a slope of $-2$ and y-intercept of $(0,2)$. Since the inequality is $\geq$, the line is solid (not dashed).

Step3: Determine shaded region

Test the origin $(0,0)$ in the inequality:
$$0 \geq -2(0) + 2 \implies 0 \geq 2$$
This is false, so we shade the region not containing the origin (above the solid line $y=-2x+2$).

Answer:

The correct graph has a solid line with slope $-2$ and y-intercept $(0,2)$, with the region above this line shaded.