8. Which of the following rational numbers lies between [tex]-\frac{5}{7}[/tex] and [tex]\frac{6}{7}[/tex]?

A. [tex]-\frac{13}{14}[/tex]
B. [tex]\frac{1}{14}[/tex]
C. [tex]\frac{7}{6}[/tex]
D. [tex]-\frac{7}{5}[/tex]



Answer :

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]