To determine which of the following options contains all of the rational numbers in the given set, we first need to identify which numbers in the set are rational.
Given set of real numbers:
[tex]\[
\left\{-\frac{15}{7}, -\sqrt{2}, \frac{2}{3}, 1, \sqrt{3}, 3.25\right\}
\][/tex]
1. Identify Rational Numbers: A rational number can be expressed as the quotient of two integers, i.e., it can be written in the form [tex]\(\frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex].
2. Check each number:
- [tex]\(-\frac{15}{7}\)[/tex]: This is a fraction with integer numerator and denominator, so it is rational.
- [tex]\(-\sqrt{2}\)[/tex]: This is the square root of a non-perfect square, so it is irrational.
- [tex]\(\frac{2}{3}\)[/tex]: This is a fraction with integer numerator and denominator, so it is rational.
- [tex]\(1\)[/tex]: This is an integer, and any integer is also a rational number (can be expressed as [tex]\(\frac{1}{1}\)[/tex]).
- [tex]\(\sqrt{3}\)[/tex]: This is the square root of a non-perfect square, so it is irrational.
- [tex]\(3.25\)[/tex]: This is a terminating decimal, which can be written as [tex]\(\frac{13}{4}\)[/tex] (a fraction with integer numerator and denominator), so it is rational.
3. List of rational numbers: From the analysis above, the rational numbers in the set are:
[tex]\[
\left\{-\frac{15}{7}, \frac{2}{3}, 1, 3.25\right\}
\][/tex]
4. Match the list with given choices:
- a.) [tex]\(\left\{-\frac{15}{7}, \frac{2}{3}, -\sqrt{2}, 1, 3.25\right\}\)[/tex]: This option includes [tex]\(-\sqrt{2}\)[/tex], which is irrational, so this is not correct.
- b.) [tex]\(\left\{-\frac{15}{7}, 1, \sqrt{3}, 3.25\right\}\)[/tex]: This option includes [tex]\(\sqrt{3}\)[/tex], which is irrational, so this is not correct.
- c.) [tex]\(\left\{-\frac{15}{7}, \frac{2}{3}, 1, 3.25\right\}\)[/tex]: This includes all and only the rational numbers we identified.
- d.) [tex]\(\left\{-\sqrt{2}, \sqrt{3}\right\}\)[/tex]: These are both irrational numbers, so this option is also not correct.
Thus, the correct choice is:
c.) [tex]\(\left\{-\frac{15}{7}, \frac{2}{3}, 1, 3.25\right\}\)[/tex]
