Question

graphing systems of linear inequalities maze
graph the given inequality. choose the ordered pair that is a solution of the system of linear inequalities.
start
graph 1: ( y > \frac{1}{2}x - 4 ), ( y geq x + 1 )
arrow to (2, 4)
graph 2: ( y leq -x + 3 ), ( y > 2x + 3 )
arrow to (-3, 5)
graph 3: ( y < \frac{1}{2}x - 3 ), ( y < -\frac{3}{2}x + 6 )
arrow to (4, 3)
graph 4: ( y geq 3x + 5 ), ( y < -\frac{1}{2}x - 2 )
arrow from graph 1: (1, -2) down to graph 5: ( y > -2x + 2 ), ( y < -\frac{1}{2}x + 5 )
arrow from graph 2: (-1, -3) down to graph 5
arrow from graph 2: (-1, 1) down to graph 6: ( y leq -\frac{1}{2}x - 1 ), ( y geq x - 1 )
arrow to (-5, -3) from graph 5 to graph 6
arrow to (2, -2) from graph 6 to graph 7: ( y geq -\frac{1}{2}x + 3 ), ( y geq \frac{1}{2}x + 1 )
arrow to (-2, 1) from graph 7 to graph 8: ( y > 2x + 5 ), ( y > \frac{1}{2}x - 2 )
arrow from graph 5: (3, 2) down to graph 9: ( y geq \frac{1}{2}x + 4 ), ( y < -x + 1 )
arrow from graph 6: (-3, 4) down to graph 9
arrow from graph 6: (4, 1) down to graph 10: ( y < x - 2 ), ( y leq -2x - 5 )
arrow to (-6, 4) from graph 9 to graph 10
arrow to (-1, -3) from graph 10 to graph 11: ( y leq \frac{1}{3}x + 2 ), ( y > -\frac{1}{3}x - 3 )
arrow from graph 7: (2, -4) down to graph 11
arrow from graph 8: (0, 6) down to graph 11
arrow from graph 8: (3, 3) down to finish

Explanation:

Step1: Verify start point (2,4)

Check if $(2,4)$ satisfies $\begin{cases}y > \frac{1}{2}x - 4 \\ y \geq x + 1\end{cases}$
- $4 > \frac{1}{2}(2)-4 = 1-4=-3$ ✔️
- $4 \geq 2+1=3$ ✔️

Step2: Move to next valid point

Test $(-1,-3)$ for $\begin{cases}y > -2x + 2 \\ y < -\frac{1}{2}x + 5\end{cases}$
- $-3 > -2(-1)+2=2+2=4$ ❌; Test $(2,4)$'s other arrow: no, use $(2,4)$ to next box? No, start's outgoing arrow is $(2,4)$ to top middle box. Wait, correct path:
Start: $(2,4)$ → top left box is start, outgoing to top middle box with $\begin{cases}y \leq -x + 3 \\ y > 2x + 3\end{cases}$. Test $(2,4)$:
- $4 \leq -2+3=1$ ❌. So start's downward arrow: $(-2,-2)$ for $\begin{cases}y > -2x + 2 \\ y < -\frac{1}{2}x + 5\end{cases}$
- $-2 > -2(-2)+2=4+2=6$ ❌. Correct first valid move: Start → $(-1,-3)$ is wrong, start's only valid outgoing is to the box with arrow $(2,4)$? No, re-express:
Correct path step-by-step:

Step1: Start at first box, test $(2,4)$

$\begin{cases}y > \frac{1}{2}x - 4 \\ y \geq x + 1\end{cases}$: $4 > 1-4=-3$, $4\geq3$ ✔️. Move to box with incoming $(2,4)$: $\begin{cases}y \leq -x + 3 \\ y > 2x + 3\end{cases}$. Test $(-3,15)$ here:
- $15 \leq 3+3=6$ ❌. Test $(6,1)$: $1 \leq -6+3=-3$ ❌. Test $(1,1)$: $1 \leq -1+3=2$ ✔️, $1>2(1)+3=5$ ❌. So back to start, take downward arrow to box with $\begin{cases}y > -2x + 2 \\ y < -\frac{1}{2}x + 5\end{cases}$, test $(-2,-2)$:
- $-2 > -2(-2)+2=6$ ❌. Test $(-5,-3)$:
- $-3 > -2(-5)+2=12$ ❌. Test $(2,4)$ here: $4> -4+2=-2$ ✔️, $4 < -\frac{1}{2}(2)+5=-1+5=4$ ❌. Test $(-1,-3)$:
- $-3 > -2(-1)+2=4$ ❌. Take downward arrow to box $\begin{cases}y \geq \frac{1}{2}x + 4 \\ y < -x + 1\end{cases}$, test $(-6,4)$:
- $4 \geq \frac{1}{2}(-6)+4=-3+4=1$ ✔️
- $4 < -(-6)+1=7$ ✔️

Step3: Move to next box with $(-3,4)$

Test $\begin{cases}y < x - 2 \\ y \leq -2x -5\end{cases}$:
- $4 < -3-2=-5$ ❌. Test $(-6,4)$'s other arrow? No, take arrow from this box to $\begin{cases}y < x - 2 \\ y \leq -2x -5\end{cases}$, test $(-1,-3)$:
- $-3 < -1-2=-3$ ❌. Test $(-2,-5)$:
- $-5 < -2-2=-4$ ✔️
- $-5 \leq -2(-2)-5=4-5=-1$ ✔️

Step4: Move to box with $\begin{cases}y \leq -\frac{1}{2}x -1 \\ y \geq x -1\end{cases}$, test $(2,-2)$:

- $-2 \leq -\frac{1}{2}(2)-1=-1-1=-2$ ✔️
- $-2 \geq 2-1=1$ ❌. Test $(-5,-3)$:
- $-3 \leq -\frac{1}{2}(-5)-1=2.5-1=1.5$ ✔️
- $-3 \geq -5-1=-6$ ✔️

Step5: Move to box $\begin{cases}y > -2x + 2 \\ y < -\frac{1}{2}x + 5\end{cases}$, test $(-1,-3)$:

- $-3 > -2(-1)+2=4$ ❌. Test $(2,4)$:
- $4> -4+2=-2$ ✔️, $4 < -1+5=4$ ❌. Take arrow to start box, test $(2,4)$ again, then take arrow to $\begin{cases}y < \frac{1}{2}x -3 \\ y < -\frac{2}{3}x +6\end{cases}$, test $(4,3)$:
- $3 < \frac{1}{2}(4)-3=2-3=-1$ ❌. Test $(2,2)$:
- $2 < 1-3=-2$ ❌. Test $(6,1)$:
- $1 < 3-3=0$ ❌. Test $(-3,15)$:
- $15 < -1.5-3=-4.5$ ❌. Back to $(-5,-3)$ → box $\begin{cases}y \leq -\frac{1}{2}x -1 \\ y \geq x -1\end{cases}$, take arrow to $\begin{cases}y \geq -\frac{2}{3}x +3 \\ y \geq \frac{1}{3}x +1\end{cases}$, test $(2,-2)$:
- $-2 \geq -\frac{4}{3}+3=\frac{5}{3}$ ❌. Test $(-2,1)$:
- $1 \geq \frac{4}{3}+3=\frac{13}{3}$ ❌. Test $(2,2)$:
- $2 \geq -\frac{4}{3}+3=\frac{5}{3}$ ✔️
- $2 \geq \frac{2}{3}+1=\frac{5}{3}$ ✔️

Step6: Move to box $\begin{cases}y < \frac{1}{2}x -3 \\ y < -\frac{2}{3}x +6\end{cases}$, test $(4,3)$:

- $3 < 2-3=-1$ ❌. Test $(6,1)$:
- $1 < 3-3=0$ ❌. Test $(-3,15)$:
- $15 < -1.5-3=-4.5$ ❌. Take arrow to $\begin{cases}y \leq -x +3 \\ y > 2x +3\end{cases}$, test $(1,1)$:
- $1 \leq -1+3=2$ ✔️, $1>2+3=5$ ❌. Test $(-3,15)$:
- $15 \leq 3+3=6$ ❌. Back to $(2,2)$ → take arrow to $\begin{cases}y \leq \frac{1}{3}x +2 \\ y > -\frac{1}{2}x -3\end{cases}$, test $(2,2)$:
- $2 \leq \frac{2}{3}+2=\frac{8}{3}$ ✔️
- $2 > -1-3=-4$ ✔️

Step7: Move to next box with $(-1,-3)$:

- $-3 \leq -\frac{1}{3}+2=\frac{5}{3}$ ✔️
- $-3 > \frac{1}{2}-3=-\frac{5}{2}$ ❌. Test $(6,-1)$:
- $-1 \leq \frac{6}{3}+2=2+2=4$ ✔️
- $-1 > -\frac{1}{2}(6)-3=-3-3=-6$ ✔️

Step8: Reach FINISH with $(6,-1)$

Answer:

The path through the maze is:
Start $
ightarrow (-5,-3)
ightarrow (-2,-5)
ightarrow (2,2)
ightarrow (6,-1)
ightarrow$ FINISH
The sequence of valid ordered pairs is $\boldsymbol{(2,4)
ightarrow (-5,-3)
ightarrow (-2,-5)
ightarrow (2,2)
ightarrow (6,-1)}$