Answer :
To compare the solution sets of the inequalities \(-23 > x\) and \(x \geq -23\), we will consider how each is represented on a number line.
1. For the inequality \(-23 > x\):
- We represent all values less than \(-23\).
- On the number line, we use an open circle at \(-23\) to indicate that \(-23\) is not included in the solution set.
- The arrow will point to the left from \(-23\), showing all numbers less than \(-23\).
2. For the inequality \(x \geq -23\):
- We represent all values greater than or equal to \(-23\).
- On the number line, we use a closed circle at \(-23\) to indicate that \(-23\) is included in the solution set.
- The arrow will point to the right from \(-23\), showing all numbers greater than or equal to \(-23\).
Now, let's examine the options given:
1. One graph uses an open circle, but the other graph uses a closed circle. Also, the rays use arrows that point in opposite directions, but they have the same endpoint.
- This option correctly describes both the use of open and closed circles and the direction of the arrows, which are opposite but emanate from the same point \(-23\).
2. They use arrows that point in opposite directions, the rays have different endpoints, and both graphs use open circles.
- This option incorrectly states that both graphs use open circles and specifies different endpoints, which does not apply since both inequations involve the same endpoint \(-23\).
3. One graph uses an open circle, but the other graph uses a closed circle. Also, the rays have different endpoints, but they use arrows that point in the same direction.
- This option wrongly states that the arrows point in the same direction and specifies different endpoints, which is not correct.
4. They use arrows that point in opposite directions, the rays have different endpoints, and both graphs use closed circles.
- This option also incorrectly states that both graphs use closed circles and different endpoints and disregards the different circle types (open vs. closed).
Considering these explanations, the most accurate description of how the graphs of the inequalities differ is:
One graph uses an open circle, but the other graph uses a closed circle. Also, the rays use arrows that point in opposite directions, but they have the same endpoint.
Thus, the correct answer is the first option.
1. For the inequality \(-23 > x\):
- We represent all values less than \(-23\).
- On the number line, we use an open circle at \(-23\) to indicate that \(-23\) is not included in the solution set.
- The arrow will point to the left from \(-23\), showing all numbers less than \(-23\).
2. For the inequality \(x \geq -23\):
- We represent all values greater than or equal to \(-23\).
- On the number line, we use a closed circle at \(-23\) to indicate that \(-23\) is included in the solution set.
- The arrow will point to the right from \(-23\), showing all numbers greater than or equal to \(-23\).
Now, let's examine the options given:
1. One graph uses an open circle, but the other graph uses a closed circle. Also, the rays use arrows that point in opposite directions, but they have the same endpoint.
- This option correctly describes both the use of open and closed circles and the direction of the arrows, which are opposite but emanate from the same point \(-23\).
2. They use arrows that point in opposite directions, the rays have different endpoints, and both graphs use open circles.
- This option incorrectly states that both graphs use open circles and specifies different endpoints, which does not apply since both inequations involve the same endpoint \(-23\).
3. One graph uses an open circle, but the other graph uses a closed circle. Also, the rays have different endpoints, but they use arrows that point in the same direction.
- This option wrongly states that the arrows point in the same direction and specifies different endpoints, which is not correct.
4. They use arrows that point in opposite directions, the rays have different endpoints, and both graphs use closed circles.
- This option also incorrectly states that both graphs use closed circles and different endpoints and disregards the different circle types (open vs. closed).
Considering these explanations, the most accurate description of how the graphs of the inequalities differ is:
One graph uses an open circle, but the other graph uses a closed circle. Also, the rays use arrows that point in opposite directions, but they have the same endpoint.
Thus, the correct answer is the first option.
