Question

4
$y < \frac{5}{2}x - 3$
$y \geq -\frac{1}{2}x + 3$

5
$y \leq 6x + 3$
$y < x - 2$

6
$3x - y > 1$
$x + y \leq 3$

Explanation:

System 4: Find intersection point

Set $\frac{5}{2}x - 3 = -\frac{1}{2}x + 3$
$$\frac{5}{2}x + \frac{1}{2}x = 3 + 3$$
$$3x = 6$$
$$x = 2$$
Substitute $x=2$ into $y = -\frac{1}{2}x + 3$:
$$y = -\frac{1}{2}(2) + 3 = 2$$

System 5: Find intersection point

Set $6x + 3 = x - 2$
$$6x - x = -2 - 3$$
$$5x = -5$$
$$x = -1$$
Substitute $x=-1$ into $y = x - 2$:
$$y = -1 - 2 = -3$$

System 6: Rewrite & find intersection

Rewrite inequalities: $y < 3x - 1$, $y \leq -x + 3$
Set $3x - 1 = -x + 3$
$$3x + x = 3 + 1$$
$$4x = 4$$
$$x = 1$$
Substitute $x=1$ into $y = -x + 3$:
$$y = -1 + 3 = 2$$

Answer:

1. For system 4: Intersection at $(2, 2)$; solution set is all points below $y=\frac{5}{2}x-3$ and above/on $y=-\frac{1}{2}x+3$
2. For system 5: Intersection at $(-1, -3)$; solution set is all points below/on $y=6x+3$ and below $y=x-2$
3. For system 6: Intersection at $(1, 2)$; solution set is all points below $y=3x-1$ and below/on $y=-x+3$