Answer:
Four rational numbers between [tex]-\frac{3}{2}[/tex] and [tex]\frac{5}{3}[/tex] are [tex]x_{1} = -\frac{13}{15}[/tex], [tex]x_{2} = -\frac{7}{30}[/tex], [tex]x_{3} = \frac{2}{5}[/tex] and [tex]x_{4} = \frac{31}{30}[/tex].
Step-by-step explanation:
First, we calculate the distance between [tex]-\frac{3}{2}[/tex] and [tex]\frac{5}{3}[/tex]:
[tex]r = \frac{5}{3}-\left(-\frac{3}{2}\right)[/tex]
[tex]r = \frac{5}{3}+\frac{3}{2}[/tex]
[tex]r = \frac{10+9}{6}[/tex]
[tex]r = \frac{19}{6}[/tex]
Then, we find four rational numbers by using the following formula:
[tex]x = -\frac{3}{2}+\left(\frac{n}{5})\cdot \left(\frac{19}{6} \right)[/tex]
First number ([tex]n = 1[/tex])
[tex]x_{1} = -\frac{13}{15}[/tex]
Second number ([tex]n = 2[/tex])
[tex]x_{2} = -\frac{7}{30}[/tex]
Third number ([tex]n = 3[/tex])
[tex]x_{3} = \frac{2}{5}[/tex]
Fourth number ([tex]n = 4[/tex])
[tex]x_{4} = \frac{31}{30}[/tex]
