Question

which of the following shows the graph of the inequality ( r geq -2.25 )?

Explanation:

Step1: Analyze the inequality

The inequality is \( r \geq -2.25 \). On a number line, this means we need to show all numbers greater than or equal to -2.25. A closed circle (filled dot) is used for "greater than or equal to" or "less than or equal to" because the endpoint is included. An open circle is for strict inequalities (greater than or less than) where the endpoint is not included.

Step2: Locate -2.25 on the number line

First, let's find where -2.25 is. The number line has marks at -4, -3.5, -3, -2.5, -2, etc. Let's see the intervals: from -2.5 to -2, the distance is 0.5, so each small tick (if any) or the midpoint? Wait, -2.25 is halfway between -2.5 and -2? Wait, -2.5 to -2 is 0.5 units. So -2.25 is 0.25 units above -2.5 (since -2.5 + 0.25 = -2.25) and 0.25 units below -2 (since -2 - 0.25 = -2.25). Wait, actually, -2.25 is between -2.5 and -2. Let's check the options:

- Option A: The arrow is going up (towards positive numbers), but the dot is at -2? Wait, no, let's look at the dots. Wait, the options:

Wait, the options are A, B, C, D. Let's re-examine:

- Option B: The arrow is going up (towards more positive numbers), and the closed circle? Wait, no, let's check the endpoints. Wait, the inequality is \( r \geq -2.25 \), so the graph should have a closed circle at -2.25 (or the correct position) and the arrow pointing to the right (towards greater numbers, i.e., more positive). Wait, maybe I misread the number line. Wait, the number line in the image: let's see the labels. The numbers are -4, -3.5, -3, -2.5, -2 (from top to bottom? Wait, no, the vertical number line? Wait, no, it's a vertical number line? Wait, the arrows are pointing up, so up is more positive (greater numbers), down is more negative (smaller numbers). Wait, that's unusual, but maybe it's a vertical number line. So the top is more positive, bottom is more negative.

Wait, let's list the options:

- Option A: Open circle? No, wait, the dot at the bottom (near -2) is a closed diamond? Wait, no, the symbols:

Wait, the problem is a vertical number line, so "up" is increasing (more positive), "down" is decreasing (more negative). So \( r \geq -2.25 \) means all r values greater than or equal to -2.25, so on the vertical number line, it's the region above (including) -2.25.

Now, let's find -2.25. The marks are at -4, -3.5, -3, -2.5, -2 (from top to bottom? Wait, no, the labels: -4 is above -3.5, which is above -3, above -2.5, above -2. So top is -4, bottom is -2? Wait, that can't be. Wait, no, maybe the number line is drawn with the top as more negative and bottom as more positive? Wait, that would be unusual, but let's check the arrows. The arrows are pointing up, so if up is more negative, then down is more positive. Wait, that's confusing. Wait, maybe it's a horizontal number line rotated 90 degrees, with left as more negative and right as more positive, but drawn vertically. So the top is left (more negative), bottom is right (more positive). Wait, that might be the case. Let's assume that: the top of the number line is more negative (left on a horizontal line), bottom is more positive (right on a horizontal line). So the arrow direction: if the arrow is pointing up, it's towards more negative; pointing down, towards more positive. Wait, no, the standard number line has right as positive, left as negative. So if this is a vertical number line, maybe up is positive (increasing) and down is negative (decreasing). So -4 is above -3.5, which is above -3, above -2.5, above -2. So -4 is more negative (smaller), -2 is more positive (larger). So the value increases as we go down the number line (since -2 is below -2.5, which is below -3, etc.). Wait, that makes sense: -4 (top) is smaller, -2 (bottom) is larger. So moving down the number line (towards the bottom) is moving towards more positive (larger) numbers.

So the inequality \( r \geq -2.25 \) means r is greater than or equal to -2.25, so we need all numbers greater than or equal to -2.25, which are the numbers below (towards the bottom) of -2.25 on this vertical number line (since bottom is larger). So the graph should have a closed circle at -2.25 (since the endpoint is included) and the arrow pointing down (towards larger numbers, i.e., more positive). Wait, but the arrows in the options:

- Option B: The arrow is pointing up (towards more negative), and the dot is at -2.25? Wait, no, let's check the endpoints. Wait, the inequality is \( r \geq -2.25 \), so the solution is all r such that r ≥ -2.25, so on the number line (vertical, with up as negative, down as positive), the graph should start at -2.25 (closed circle) and go down (towards positive numbers). Wait, maybe I got the direction wrong. Let's take a step back.

Alternative approach: Let's convert the vertical number line to horizontal for easier understanding. Imagine rotating the image 90 degrees clockwise, so the number line is horizontal, with left as negative and right as positive. Then:

- The top of the original image (with -4) becomes the left end of the horizontal line.
- The bottom (with -2) becomes the right end.

So the horizontal number line would have -4 on the left, -3.5, -3, -2.5, -2 on the right. So moving from left to right (top to bottom in the original) is increasing (from -4 to -2, which is increasing). So the inequality \( r \geq -2.25 \) means r is greater than or equal to -2.25, so on the horizontal number line (left to right), we need a closed circle at -2.25 (since it's included) and the arrow pointing to the right (towards larger numbers, i.e., towards -2, -1, etc.).

Now, let's find -2.25 on this horizontal line. The points are at -4 (left), -3.5, -3, -2.5, -2 (right). The distance between -2.5 and -2 is 0.5 units (since -2.5 + 0.5 = -2). So -2.25 is 0.25 units to the right of -2.5 (because -2.5 + 0.25 = -2.25) and 0.25 units to the left of -2 (because -2 - 0.25 = -2.25). So -2.25 is the midpoint between -2.5 and -2? Wait, no, 0.5 units between -2.5 and -2, so each 0.25 units would be a tick. So -2.5, -2.25, -2. So -2.25 is between -2.5 and -2, closer to -2.5? Wait, no, -2.5 to -2 is 0.5, so half of that is 0.25, so -2.5 + 0.25 = -2.25, so it's 0.25 above -2.5 (towards -2).

Now, let's check the options:

- Option B: The arrow is pointing to the right (towards -2, -1, etc.), and the closed circle? Wait, in the original image, option B has a closed circle (filled dot) at the top (more negative) and the arrow going down (towards positive). Wait, no, let's look at the symbols:

Wait, the problem is a vertical number line, so:

- If the arrow is going up, it's towards more negative (smaller numbers).
- If the arrow is going down, it's towards more positive (larger numbers).

The inequality \( r \geq -2.25 \) means r is greater than or equal to -2.25, so we need all numbers greater than or equal to -2.25, which are the numbers larger than -2.25 (more positive). So the graph should have a closed circle at -2.25 (since the endpoint is included) and the arrow pointing down (towards more positive numbers, i.e., larger r).

Now, let's find -2.25 on the vertical number line. The marks are at -4 (top), -3.5, -3, -2.5, -2 (bottom). So -2.25 is between -2.5 and -2 (since -2.5 > -2.25 > -2). So the closed circle should be at -2.25, and the arrow pointing down (towards -2, which is larger than -2.25).

Now, let's check the options:

- Option B: The closed circle is at -2.25? Wait, no, let's see the dots. Wait, option B has a closed circle (filled dot) at the top (near -4?) No, wait, the labels:

Wait, the original image:

- Option A: The dot is at -2 (bottom), open or closed? It's a diamond, maybe closed? The arrow is up.
- Option B: The dot is at -2.25? Wait, no, the numbers: -4, -3.5, -3, -2.5, -2. Wait, -2.25 is between -2.5 and -2. Let's see the length of the blue line. Option B's blue line starts at a closed circle (filled dot) and goes down (towards -2). Wait, maybe I made a mistake. Let's think again.

The inequality is \( r \geq -2.25 \). So all real numbers r that are greater than or equal to -2.25. On a number line, this is represented by a closed circle at -2.25 (because the inequality includes -2.25) and an arrow pointing to the right (towards positive infinity, i.e., larger numbers). Now, if the number line is vertical, with up as positive (larger numbers) and down as negative (smaller numbers), then the arrow should point up, and the closed circle at -2.25. Wait, that makes more sense. So up is larger (more positive), down is smaller (more negative). So -4 is down (smaller), -2 is up (larger). Wait, no, that's the opposite. Wait, standard number line: right is positive (larger), left is negative (smaller). So a vertical number line: up is right (positive, larger), down is left (negative, smaller). So -4 is down (smaller), -2 is up (larger). So the number line has -4 at the bottom, -3.5 above it, -3 above, -2.5 above, -2 at the top. Wait, that would make sense: up is more positive (larger), so -2 is at the top (larger), -4 at the bottom (smaller). So in that case, \( r \geq -2.25 \) means r is greater than or equal to -2.25, so the graph should have a closed circle at -2.25 (which is between -2.5 and -2, closer to -2.5) and the arrow pointing up (towards -2, which is larger than -2.25).

Now, let's check the options:

- Option B: The blue line starts at a closed circle (filled dot) and goes up (towards -2). The position of the dot: between -2.5 and -2, which is -2.25. So that's correct. Because -2.25 is between -2.5 and -2, and the closed circle is at -2.25, with the arrow pointing up (towards larger numbers, i.e., r ≥ -2.25).

Other options:

- Option A: The dot is at -2, arrow up. But -2 is greater than -2.25, but the inequality is r ≥ -2.25, so the endpoint should be at -2.25, not -2.

- Option C: The dot is at -2, arrow down. That's r ≤ -2, which is not our inequality.

- Option D: Open circle at -2, arrow down. That's r < -2, which is not our inequality.

So the correct option is B, because it has a closed circle at -2.25 (the correct endpoint) and the arrow pointing up (towards larger numbers, indicating r ≥ -2.25).

Answer:

B) (the option with the closed circle at -2.25 and the arrow pointing up)