To solve the problem of comparing [tex]\(-3.\overline{5}\)[/tex] and [tex]\(-\frac{10}{3}\)[/tex], let's break the problem down step by step.
First, we need to represent [tex]\( -3.\overline{5} \)[/tex] in a more precise form. [tex]\( -3.\overline{5} \)[/tex] is a repeating decimal, meaning it extends infinitely as follows:
[tex]\[ -3.55555\ldots \][/tex]
In order to facilitate comparison, it can sometimes be helpful to think of repeating decimals with a few more decimal places. Here [tex]\(-3.\overline{5}\)[/tex] can be approximated more clearly as:
[tex]\[ -3.55555\ldots \][/tex]
Next, we'll convert the fraction [tex]\( -\frac{10}{3} \)[/tex] into its decimal form to compare.
[tex]\[ -\frac{10}{3} = -3.33333\ldots \][/tex]
Now that we have both numbers in decimal form, we can directly compare:
[tex]\[ -3.55555\ldots \][/tex]
[tex]\[ -3.33333\ldots \][/tex]
When comparing these two numbers, it's clear that [tex]\( -3.55555\ldots \)[/tex] is less than [tex]\( -3.33333\ldots \)[/tex], because as we move farther from zero on the number line in the negative direction, the value of the number decreases. Thus:
[tex]\[ -3.\overline{5} < -\frac{10}{3} \][/tex]
Therefore, the correct comparison is:
\[ -3 . \overline{5} < -\frac{10}{3} \.]
So, the correct selection from the given options is:
\[ -3 \overline{5} < -\frac{10}{3} \.]
