Sure, I'd be happy to help you find rational numbers between the given pairs.
(a) For the range between [tex]\(-5.6\)[/tex] and [tex]\(-5.7\)[/tex]:
To find three rational numbers between [tex]\(-5.6\)[/tex] and [tex]\(-5.7\)[/tex], we need to identify values that lie numerically between these two points. Here are three such rational numbers:
- [tex]\(-5.69\)[/tex] is between [tex]\(-5.7\)[/tex] and [tex]\(-5.6\)[/tex]
- [tex]\(-5.65\)[/tex] is between [tex]\(-5.7\)[/tex] and [tex]\(-5.6\)[/tex]
- [tex]\(-5.61\)[/tex] is between [tex]\(-5.7\)[/tex] and [tex]\(-5.6\)[/tex]
Therefore, the three rational numbers between [tex]\(-5.7\)[/tex] and [tex]\(-5.6\)[/tex] are:
[tex]\[ -5.69, -5.65, -5.61 \][/tex]
(b) For the range between [tex]\(\frac{4}{6}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex]:
To find three rational numbers between [tex]\(\frac{4}{6}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex], it can be helpful to convert these fractions into decimal form or find numbers that sit in this range by considering equivalent fractions. Here are three such rational numbers:
- [tex]\(\frac{4.2}{6}\)[/tex] is between [tex]\(\frac{4}{6}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex]
- [tex]\(\frac{4.4}{6}\)[/tex] is between [tex]\(\frac{4}{6}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex]
- [tex]\(\frac{4.6}{6}\)[/tex] is between [tex]\(\frac{4}{6}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex]
To express these as simpler fractions:
- [tex]\(\frac{4.2}{6} = \frac{21}{30} = 0.7000000000000001\)[/tex]
- [tex]\(\frac{4.4}{6} = \frac{22}{30} = 0.7333333333333334\)[/tex]
- [tex]\(\frac{4.6}{6} = \frac{23}{30} = 0.7666666666666666\)[/tex]
Thus, the three rational numbers between [tex]\(\frac{4}{6}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex] are:
[tex]\[ \frac{21}{30}, \frac{22}{30}, \frac{23}{30} \][/tex]
Summarizing:
(a) The rational numbers between [tex]\(-5.7\)[/tex] and [tex]\(-5.6\)[/tex] are:
[tex]\[ -5.69, -5.65, -5.61 \][/tex]
(b) The rational numbers between [tex]\(\frac{4}{6}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex] are:
[tex]\[ \frac{21}{30}, \frac{22}{30}, \frac{23}{30} \][/tex]
