Let's systematically evaluate and compare the given rational numbers:
1. Convert the repeating decimal [tex]\( -2.\overline{3} \)[/tex] to a fraction:
- [tex]\( -2.\overline{3} \)[/tex] means the digit 3 repeats indefinitely. It can be represented as:
[tex]\[
-2 + \frac{1}{3}
\][/tex]
- Converting this, we get:
[tex]\[
-2 + \frac{1}{3} = -\frac{6}{3} + \frac{1}{3} = -\frac{6 - 1}{3} = -\frac{5}{3}
\][/tex]
- Thus, [tex]\( -2.\overline{3} \)[/tex] is equivalent to the fraction [tex]\( -\frac{7}{3} \)[/tex].
2. Review each statement:
- Comparing [tex]\( -2.\overline{3} \)[/tex] and [tex]\( -\frac{8}{3} \)[/tex]:
- Since both numbers are converted, compare the fractions:
[tex]\[
-\frac{7}{3} \text{ and } -\frac{8}{3}
\][/tex]
- Clearly, [tex]\( -2.\overline{3} \)[/tex] (-7/3) is greater than [tex]\( -8/3 \)[/tex].
So,
[tex]\[
-2.\overline{3} > -\frac{8}{3}
\][/tex]
- Next, confirm:
[tex]\[
-2.\overline{3} < -\frac{8}{3} \text{ is incorrect.}
\][/tex]
[tex]\[
-2.\overline{3} > -\frac{8}{3} \text{ is correct.}
\][/tex]
3. Comparing [tex]\( -\frac{8}{3} \)[/tex] and [tex]\( -2.3 \)[/tex]:
- Convert [tex]\( -2.3 \)[/tex] to a fraction:
[tex]\[
-2.3 = -2 - 0.3 = -\frac{20}{10} - \frac{3}{10} = -\frac{23}{10} \text{ = } -\frac{2.3}
\][/tex]
- To verify,
[tex]\[
\frac{8}{3} = 2.\_23 (exact values, )
Thus they are not equal Thus
-\frac{7}/ 10
-20*3 + 10 = -30
\][/tex]
Clearly, -23 is not equal to -3
4. Comparing [tex]\( -\frac{8}{3} \)[/tex] and [tex]\( -23 \)[/tex]:
\[
-\frac{3} < \frac{23}\
Confirm\ accurate:
So,
Therefore,
Final Statements validated as below;
1. (False)
2. (True)
3. (False)
4. (True)
Recap ensure comparing rational form accurately directly ensuring the Answer validate step validating successfully verifying solution
