To solve the problem of identifying which rational number lies between [tex]\(-\frac{5}{7}\)[/tex] and [tex]\(\frac{6}{7}\)[/tex], we need to evaluate each of the given options to see if they fall within this range.
1. Option (a): [tex]\(-\frac{13}{14}\)[/tex]
[tex]\(\frac{13}{14} \approx 0.9286\)[/tex]
Therefore:
[tex]\(-\frac{13}{14} \approx -0.9286\)[/tex]
Comparing this to the range:
- [tex]\( -\frac{13}{14} < -\frac{5}{7} \)[/tex]
Therefore, [tex]\(-\frac{13}{14}\)[/tex] is not in the range [tex]\([- \frac{5}{7}, \frac{6}{7}]\)[/tex].
2. Option (b): [tex]\(\frac{1}{14}\)[/tex]
[tex]\(\frac{1}{14} \approx 0.0714\)[/tex]
Comparing this to the range:
- [tex]\( -\frac{5}{7} \approx -0.714 \)[/tex]
- [tex]\( \frac{6}{7} \approx 0.857 \)[/tex]
So:
- [tex]\( -0.714 < 0.0714 < 0.857 \)[/tex]
Therefore, [tex]\(\frac{1}{14}\)[/tex] is within the range [tex]\([- \frac{5}{7}, \frac{6}{7}]\)[/tex].
3. Option (c): [tex]\(\frac{7}{6}\)[/tex]
[tex]\(\frac{7}{6} \approx 1.1667\)[/tex]
Comparing this to the range:
- [tex]\( \frac{7}{6} > \frac{6}{7} \)[/tex]
Therefore, [tex]\(\frac{7}{6}\)[/tex] is not in the range [tex]\([- \frac{5}{7}, \frac{6}{7}]\)[/tex].
4. Option (d): [tex]\(-\frac{7}{5}\)[/tex]
[tex]\(\frac{7}{5} = 1.4\)[/tex]
Therefore:
[tex]\(-\frac{7}{5} = -1.4\)[/tex]
Comparing this to the range:
- [tex]\( -\frac{7}{5} < -\frac{5}{7} \)[/tex]
Therefore, [tex]\(-\frac{7}{5}\)[/tex] is not in the range [tex]\([- \frac{5}{7}, \frac{6}{7}]\)[/tex].
Upon carefully evaluating all options, we find that the rational number that lies between [tex]\(-\frac{5}{7}\)[/tex] and [tex]\(\frac{6}{7}\)[/tex] is:
(b) [tex]\(\frac{1}{14}\)[/tex]
