Question

1. $3x - 2y > -2$

$x - 2y \geq 2$

Explanation:

Step1: Rewrite inequalities to slope-intercept form

For $3x - 2y > -2$:
$-2y > -3x - 2$
$y < \frac{3}{2}x + 1$

For $x - 2y \geq 2$:
$-2y \geq -x + 2$
$y \leq \frac{1}{2}x - 1$

Step2: Identify boundary lines

• For $y < \frac{3}{2}x + 1$: Boundary is $y = \frac{3}{2}x + 1$ (dashed line, since inequality is strict).
• x-intercept: Set $y=0$, $0 = \frac{3}{2}x + 1 \implies x = -\frac{2}{3}$
• y-intercept: Set $x=0$, $y=1$
• For $y \leq \frac{1}{2}x - 1$: Boundary is $y = \frac{1}{2}x - 1$ (solid line, since inequality is non-strict).
• x-intercept: Set $y=0$, $0 = \frac{1}{2}x - 1 \implies x=2$
• y-intercept: Set $x=0$, $y=-1$

Step3: Determine solution regions

• For $y < \frac{3}{2}x + 1$: Shade below the dashed line.
• For $y \leq \frac{1}{2}x - 1$: Shade below the solid line.
• The overlapping shaded area is the solution to the system.

Step4: Find intersection of boundaries

Set $\frac{3}{2}x + 1 = \frac{1}{2}x - 1$
$\frac{3}{2}x - \frac{1}{2}x = -1 - 1$
$x = -2$
Substitute $x=-2$ into $y = \frac{1}{2}x - 1$:
$y = \frac{1}{2}(-2) - 1 = -2$
Intersection point: $(-2, -2)$

Answer:

The solution is the overlapping region below the dashed line $y = \frac{3}{2}x + 1$ and below the solid line $y = \frac{1}{2}x - 1$, bounded by the intersection point $(-2, -2)$.
To graph:

1. Draw dashed line through $(-\frac{2}{3}, 0)$ and $(0, 1)$; shade below it.
2. Draw solid line through $(2, 0)$ and $(0, -1)$; shade below it.
3. The overlapping shaded area is the solution set.