Question

consider the system of inequalities,
$-x + 4y \geq -21$
$x + 3y < -7$
enter three different points, separated by commas, that are solutions to the system of inequalities.

enter three different points, separated by commas, that are not solutions to the system of inequalities.

Explanation:

Step1: Rewrite inequalities for clarity

First inequality: $-x + 4y \geq -21 \implies x \leq 4y + 21$
Second inequality: $x + 3y < -7 \implies x < -3y -7$

Step2: Find valid solution points

Pick $y=-3$:
For second inequality: $x < -3(-3)-7 = 9-7=2$
For first inequality: $x \leq 4(-3)+21 = -12+21=9$
Choose $x=0$: $(0,-3)$ satisfies both.

Pick $y=-4$:
Second inequality: $x < -3(-4)-7=12-7=5$
First inequality: $x \leq 4(-4)+21=-16+21=5$
Choose $x=4$: $(4,-4)$ satisfies both.

Pick $y=-2$:
Second inequality: $x < -3(-2)-7=6-7=-1$
First inequality: $x \leq 4(-2)+21=-8+21=13$
Choose $x=-2$: $(-2,-2)$ satisfies both.

Step3: Find non-solution points

Point violating second inequality: $(0,0)$: $0+0=0
less -7$
Point violating first inequality: $(20,-3)$: $-20 + 4(-3)=-32 < -21$
Point violating both: $(10,0)$: $-10+0=-10
geq -21$ and $10+0=10
less -7$

Answer:

Solution points: $(0,-3), (4,-4), (-2,-2)$
Non-solution points: $(0,0), (20,-3), (10,0)$