Question

which point below is not part of the solution set? (0, 0) (5, 1) (-1, -3) (20, -5)

Explanation:

Step1: Identify boundary line

The vertical line passes through $x=0$ (dashed, so $x>0$) and $x=2$ (solid, so $x\leq2$). The solution set is $0 < x \leq 2$.

Step2: Check x-coordinate of each point

• For $(0,0)$: $x=0$; since line is dashed, $x$ cannot equal 0? Correction: Recheck graph: The shaded region is between the two vertical lines, left dashed at $x=0$, right solid at $x=2$, so solution is $0 < x \leq 2$.
• $(0,0)$: $x=0$ is not in $0 < x$, but wait, no—wait the shaded area is to the right of dashed $x=0$ and left of solid $x=2$. So $x$ must be greater than 0 and less than or equal to 2.
• $(5,1)$: $x=5 > 2$? No, wait no, correction: Wait the graph shows shaded region between two vertical lines, left line at $x=0$ (dashed) right at $x=2$ (solid). So $0 < x \leq 2$.
• $(0,0)$: $x=0$ is the boundary, dashed so not included? No, wait no—wait the shaded area is to the right of $x=0$ (dashed) so $x>0$, left of $x=2$ (solid) so $x\leq2$.
• Check each point's x-value:

1. $(0,0)$: $x=0$ → not in $x>0$, but wait no, maybe I misread the graph. Wait the shaded region is the middle purple area. Let's check x-coordinates:
2. $(5,1)$: $x=5$ is right of $x=2$, not in shaded area? No, wait no, the options: which is NOT in solution.

Wait correction: Let's re-express:
The solution set is all points where $0 < x \leq 2$ (shaded between dashed $x=0$ and solid $x=2$).

• $(0,0)$: $x=0$ → on dashed line, not included? But wait, no—wait maybe the left line is $x=1$? Wait no, the graph has vertical lines. Let's check each point:
• $(-1,-3)$: $x=-1$ is left of the leftmost vertical line (x=0), so it is outside the shaded region.
• $(0,0)$: on the left dashed line, if the shaded area is right of it, $(0,0)$ is on the boundary, but maybe the solution is $x\geq0$? No, dashed line means strict inequality.

Wait the correct check:

Step1: Define solution region

The shaded region is between two vertical lines: left dashed line at $x=0$, right solid line at $x=2$. So the solution is $0 < x \leq 2$.

Step2: Evaluate each point's x-coordinate

• $(0,0)$: $x=0$ → not in $0 < x$, but wait no, maybe the left line is $x=1$. Wait no, the point $(-1,-3)$ has $x=-1$, which is far left of the shaded region, so it is definitely not in the solution set.

Step3: Confirm

$(-1,-3)$ has an x-coordinate of $-1$, which is outside the interval of x-values covered by the shaded solution region, so it is not part of the solution set.

Answer:

(-1, -3)