Question

the following graph represents an inequality. select the ordered pairs which are solutions to it. draw (graph with grid, axes, and blue shaded region)
□ (-2,2)
□ (4,-14)
□ (2,0)
□ (-3,14)
□ (1,-2)
□ (0,0)

Explanation:

Step1: Analyze the Graph

The graph shows a line (probably a linear inequality) with a shaded region. We need to check which ordered pairs lie within the shaded area. First, let's assume the line equation (from the graph, it seems to pass through (0,0) and has a steep slope, maybe \( y = -5x \) or similar, but visually, the shaded region is to the right of the line (or based on the grid, let's check each point's position relative to the shaded area).

Step2: Check Each Ordered Pair

- **\((-2, 2)\)**: Let's see the x-coordinate is -2. The shaded region (from the graph) seems to be where x is positive? Wait, no, the grid has negative x on the left. Wait, the graph's x-axis: left is negative (15,10,5,0,5,10,15? Wait, maybe the x-axis is labeled with negative on the left? Wait, the graph has x from -15 to 15? Wait, the labels: 15,10,5,0,5,10,15 – so left of 0 is negative (15,10,5,0, then 5,10,15 on the right? Wait, no, standard graph: left is negative, right is positive. So the shaded region is to the right of the blue line? Wait, the blue line is a steep line, maybe with slope -5 (since from (0,0) going up, but the shaded area is below or above? Wait, the shaded area is blue, let's check the points:

- \((-2, 2)\): x=-2 (left of y-axis), y=2. Is this in the shaded area? The graph's shaded area – looking at the grid, the blue region is on the right side (positive x) and maybe lower? Wait, no, the line is steep, maybe the inequality is \( y < -5x \) or \( y > -5x \)? Wait, let's check the point (4, -14): x=4, y=-14. Let's see if that's in the shaded area.

Wait, maybe a better approach: the shaded region is where the points are. Let's list the points:

1. \((-2, 2)\): x=-2, y=2. The shaded area (from the graph) – the blue region is on the right (positive x) and lower? Wait, no, the graph's blue area is on the right side (x positive) and maybe below the line? Wait, the line goes from top left to bottom right, passing through (0,0). So the shaded region is to the right of the line (since the line is at x= something? Wait, maybe the line is \( x = -y/5 \) or similar. Wait, maybe the line has equation \( y = -5x \). Let's check:

For \( y = -5x \), at x=1, y=-5; x=2, y=-10; x=3, y=-15; x=-1, y=5; x=-2, y=10.

Now, the shaded region: if the line is \( y = -5x \), and the shaded area is below the line (since the line is steep, and the shaded area is blue, maybe below). Wait, let's check the points:

- \((-2, 2)\): Plug into \( y < -5x \)? \( 2 < -5*(-2) = 10 \). 2 < 10: true. But is this in the shaded area? Wait, the graph's shaded area – looking at the grid, the blue region is on the right (positive x) and lower. Wait, maybe I got the x-axis reversed. Wait, the labels on the x-axis: left side (above the x-axis) has 15,10,5,0, and right side has 5,10,15. So left of 0 is negative (15,10,5,0), right is positive (5,10,15). So the blue shaded area is on the right side (positive x) and below the line? Wait, no, the point (4, -14): x=4 (positive), y=-14 (negative). Let's see if that's in the shaded area.

Wait, maybe the line is \( y = -5x \), and the inequality is \( y < -5x \) (since the shaded area is below the line). Let's check each point:

- \((-2, 2)\): \( 2 < -5*(-2) = 10 \) → 2 < 10: true. But x=-2 (left of y-axis). Is the shaded area including left? The graph's blue area – looking at the graph, the blue region is on the right (positive x) and lower. Wait, maybe the line is \( x = -y/5 \), but this is getting confusing. Alternatively, visually, the shaded area is where x is positive (right of y-axis) and y is negative or low? Wait, the point (4, -14): x=4 (right), y=-14 (low). (2,0): x=2 (right), y=0. (1, -2): x=1 (right), y=-2. (0,0): on the line. (-3,14): x=-3 (left), y=14 (high). (-2,2): x=-2 (left), y=2 (medium).

Wait, maybe the shaded area is to the right of the blue line (positive x) and below the line. Let's check the points:

- \((4, -14)\): x=4 (right), y=-14. Is this in the shaded area? The blue region is on the right, so yes.
- \((2, 0)\): x=2 (right), y=0. In the shaded area?
- \((1, -2)\): x=1 (right), y=-2. In the shaded area?
- \((0, 0)\): on the line (boundary, if the line is dashed or solid? The line looks solid? Wait, the problem says "inequality" – if the line is solid, then points on the line are included. But (0,0) is on the line. Wait, but let's check the graph again.

Wait, maybe the correct points are those with positive x (since the shaded area is on the right side, x > 0? Wait, no, the x-axis labels: 15,10,5,0,5,10,15 – so left of 0 is negative (15,10,5,0), right is positive (5,10,15). So the shaded area is on the right (positive x) side of the blue line. Let's check each point:

1. \((-2, 2)\): x=-2 (negative) → not in shaded area (shaded is right of line, which is on positive x? No, the line is steep, maybe crossing the y-axis at (0,0) and going up to the left and down to the right. So the shaded area is below the line (since the line is steep, going from top left to bottom right). So for a point (x,y), if y < -5x (assuming slope -5), let's check:

- \((-2, 2)\): y=2, -5x=10 → 2 < 10 → true. But x=-2 (left of y-axis). Is this in the shaded area? The graph's shaded area – looking at the blue region, it's on the right (positive x) and lower. Wait, maybe the line is \( x = -y/5 \), but this is unclear. Alternatively, let's look at the grid:

The blue shaded area is on the right side (x ≥ 0? No, x=4 is in the shaded area, x=2, x=1, x=0. Let's check the points:

- \((4, -14)\): x=4 (right), y=-14 (low) → in shaded area.
- \((2, 0)\): x=2 (right), y=0 → in shaded area?
- \((1, -2)\): x=1 (right), y=-2 → in shaded area?
- \((0, 0)\): on the line → if the line is solid, included.
- \((-3, 14)\): x=-3 (left), y=14 (high) → not in shaded area.
- \((-2, 2)\): x=-2 (left), y=2 → not in shaded area (shaded is right).

Wait, maybe I made a mistake. Let's re-express the graph: the blue line is a vertical line? No, it's a steep line. Wait, the graph's x-axis: the labels are 15,10,5,0,5,10,15 – so the x-coordinate increases to the right (standard). The y-axis: 15,10,5,0,-5 (wait, the y-axis has 15,10,5,0, then -5? The graph shows a blue line going from top left (x negative, y positive) to bottom right (x positive, y negative), passing through (0,0). The shaded area is below the line (since the blue is below the line? Or above? Wait, the blue area is the solution region. Let's check the point (4, -14): x=4, y=-14. Is this below the line? The line at x=4: y would be -5*4 = -20? No, maybe the slope is -5, so equation y = -5x. Then at x=4, y=-20. So (4, -14) is above y=-20, but is that in the shaded area? Wait, maybe the inequality is y > -5x. Then (4, -14): -14 > -20 → true. (2,0): 0 > -10 → true. (1, -2): -2 > -5 → true. (0,0): 0 > 0? No, equal. (-2,2): 2 > 10? No. (-3,14):14 > 15? No. (4, -14): yes. (2,0): yes. (1, -2): yes. (0,0): no (since 0 is not greater than 0). Wait, but the line is solid, so maybe ≥. Then (0,0) would be included. But let's check the graph again.

Alternatively, maybe the line is x = -y/5, but this is too time-consuming. Let's use the visual: the shaded area is on the right side (positive x) and includes points with positive x and low y. Let's check each point:

- \((-2, 2)\): x=-2 (left) → no.
- \((4, -14)\): x=4 (right), y=-14 (low) → yes.
- \((2, 0)\): x=2 (right), y=0 → yes.
- \((-3, 14)\): x=-3 (left) → no.
- \((1, -2)\): x=1 (right), y=-2 → yes.
- \((0, 0)\): on the line → if the line is solid, yes. But let's see the options.

Wait, maybe the correct points are (4, -14), (2, 0), (1, -2), and (0, 0)? Wait, no, let's check the graph again. The blue shaded area: looking at the grid, the blue region is to the right of the blue line (which is a steep line) and below it? Wait, the line is from (0,0) going up to the left (x negative, y positive) and down to the right (x positive, y negative). So the shaded area is below the line (y < -5x) or above? Let's take (4, -14): y=-14, -5x=-20 → -14 > -20 → so y > -5x. So the inequality is y ≥ -5x (if the line is solid). Then:

- \((-2, 2)\): y=2, -5x=10 → 2 ≥ 10? No.
- \((4, -14)\): -14 ≥ -20 → yes.
- \((2, 0)\): 0 ≥ -10 → yes.
- \((-3, 14)\):14 ≥ 15? No.
- \((1, -2)\): -2 ≥ -5 → yes.
- \((0, 0)\):0 ≥ 0 → yes.

But maybe the line is dashed, so (0,0) is not included. But the graph's line looks solid. Alternatively, maybe the correct points are (4, -14), (2, 0), (1, -2), and (0, 0). But let's check the options again.

Wait, maybe I made a mistake. Let's look at the x and y coordinates:

- The shaded area is on the right side (x ≥ 0) and below the line. Let's check each point:

1. \((-2, 2)\): x=-2 (left) → no.
2. \((4, -14)\): x=4 (right), y=-14 (low) → yes.
3. \((2, 0)\): x=2 (right), y=0 → yes.
4. \((-3, 14)\): x=-3 (left) → no.
5. \((1, -2)\): x=1 (right), y=-2 → yes.
6. \((0, 0)\): on the line → yes (if solid).

But maybe the answer is (4, -14), (2, 0), (1, -2), and (0, 0). But let's confirm with the graph's shaded area. The blue region is on the right, so x positive, and y values that are lower (more negative) as x increases. So (4, -14) is in, (2,0) is in, (1,-2) is in, (0,0) is on the line. (-2,2) and (-3,14) are on the left (x negative) so not in.

Answer:

The ordered pairs that are solutions are \((4, -14)\), \((2, 0)\), \((1, -2)\), and \((0, 0)\) (and maybe \((-2, 2)\) if the shaded area includes left, but based on the graph's visual, likely the right side). Wait, maybe I messed up the x-axis direction. Wait, the x-axis labels: 15,10,5,0,5,10,15 – so left of 0 is negative (15,10,5,0), right is positive (5,10,15). So the shaded area is on the right (positive x) side. So the correct points are:

\((4, -14)\), \((2, 0)\), \((1, -2)\), \((0, 0)\) (and maybe \((-2, 2)\) if the line is on the left, but no, the line is steep, so left of the line is x < -y/5, but this is confusing. Alternatively, the correct points are:

\((4, -14)\), \((2, 0)\), \((1, -2)\), and \((0, 0)\). But let's check again:

• \((-2, 2)\): x=-2 (left), y=2 – the shaded area is on the right, so no.
• \((-3, 14)\): x=-3 (left), y=14 – no.
• \((4, -14)\): x=4 (right), y=-14 – yes.
• \((2, 0)\): x=2 (right), y=0 – yes.
• \((1, -2)\): x=1 (right), y=-2 – yes.
• \((0, 0)\): on the line – yes (if solid).

So the solutions are \((4, -14)\), \((2, 0)\), \((1, -2)\), and \((0, 0)\). But maybe the line is dashed, so (0,0) is not included. But the graph's line looks solid. Alternatively, maybe the correct answer is (4, -14), (2, 0), (1, -2), and (0, 0).