To determine on which number line the values [tex]\(-2\)[/tex], [tex]\(-1\)[/tex], and [tex]\(\frac{1}{2}\)[/tex] are correctly placed, we need to examine the order of numbers on each line to check if they are in the correct ascending order.
Let's look at the first number line:
[tex]\[
\begin{array}{lllll}
-2 & -1 & \frac{1}{2} & 0 & 1
\end{array}
\][/tex]
In this arrangement:
- The numbers begin with [tex]\(-2\)[/tex], which is correctly positioned as the smallest number.
- Next is [tex]\(-1\)[/tex], which is greater than [tex]\(-2\)[/tex] but less than [tex]\(\frac{1}{2}\)[/tex].
- Then [tex]\(\frac{1}{2}\)[/tex], which is larger than [tex]\(-1\)[/tex] but should be less than [tex]\(0\)[/tex].
- However, [tex]\(0\)[/tex] comes after [tex]\(\frac{1}{2}\)[/tex], but [tex]\(0\)[/tex] should be less than [tex]\(\frac{1}{2}\)[/tex]. This makes the order incorrect.
- Finally, [tex]\(1\)[/tex] is correctly positioned as the largest number in this sequence.
Thus, the first number line is incorrect because [tex]\(\frac{1}{2}\)[/tex] should be placed after [tex]\(0\)[/tex].
Next, let's examine the second number line:
[tex]\[
\begin{array}{llllll}
-2 & -1 & 0 & \frac{1}{2} & 1
\end{array}
\][/tex]
In this arrangement:
- It starts with [tex]\(-2\)[/tex], which is correct as the smallest number.
- Next is [tex]\(-1\)[/tex], which is correctly positioned between [tex]\(-2\)[/tex] and [tex]\(0\)[/tex].
- Following [tex]\(-1\)[/tex] comes [tex]\(0\)[/tex], which is correctly placed.
- After [tex]\(0\)[/tex], we have [tex]\(\frac{1}{2}\)[/tex], which is greater than [tex]\(0\)[/tex].
- Finally, [tex]\(1\)[/tex] is the largest number, thus correctly placed at the end.
The numbers on the second number line are in the correct ascending order:
[tex]\[
-2 < -1 < 0 < \frac{1}{2} < 1
\][/tex]
Therefore, the second number line correctly locates the values [tex]\(\frac{1}{2}\)[/tex], [tex]\(-1\)[/tex], and [tex]\(-2\)[/tex]. The correct number line is:
[tex]\[
\begin{array}{llllll}
-2 & -1 & 0 & \frac{1}{2} & 1
\end{array}
\][/tex]
