Does the closure property for addition hold for the following pairs of rational numbers? a) 2/3 and 6 b) 1/8 and 3/4 c) -2/7 and 3/4

The numbers are $\frac{2}{3}$ and 6. Both are rational numbers.

$$\frac{2}{3} + 6 = \frac{2}{3} + \frac{18}{3} = \frac{2+18}{3} = \frac{20}{3}$$

The sum, $\frac{20}{3}$, is a rational number.
The closure property for addition holds for rational numbers.

b)

The numbers are $\frac{1}{8}$ and $\frac{3}{4}$. Both are rational numbers.

$$\frac{1}{8} + \frac{3}{4} = \frac{1}{8} + \frac{3 \times 2}{4 \times 2} = \frac{1}{8} + \frac{6}{8} = \frac{1+6}{8} = \frac{7}{8}$$

The sum, $\frac{7}{8}$, is a rational number.
The closure property for addition holds for rational numbers.

c)

The numbers are $-\frac{2}{7}$ and $\frac{3}{4}$. Both are rational numbers.

$$-\frac{2}{7} + \frac{3}{4} = -\frac{2 \times 4}{7 \times 4} + \frac{3 \times 7}{4 \times 7} = -\frac{8}{28} + \frac{21}{28} = \frac{-8+21}{28} = \frac{13}{28}$$

The sum, $\frac{13}{28}$, is a rational number.
The closure property for addition holds for rational numbers.

d)

The numbers are $-\frac{1}{7}$ and $-\frac{1}{3}$. Both are rational numbers.

$$-\frac{1}{7} + \left(-\frac{1}{3}\right) = -\frac{1}{7} - \frac{1}{3} = -\frac{1 \times 3}{7 \times 3} - \frac{1 \times 7}{3 \times 7} = -\frac{3}{21} - \frac{7}{21} = \frac{-3-7}{21} = -\frac{10}{21}$$

The sum, $-\frac{10}{21}$, is a rational number.
The closure property for addition holds for rational numbers.

In all cases, the sum of the two given rational numbers is also a rational number. Therefore, the closure property for addition holds for all these pairs within the set of rational numbers.

a) Sum: $\boxed{\frac{20}{3}}$. The closure property holds.

b) Sum: $\boxed{\frac{7}{8}}$. The closure property holds.

c) Sum: $\boxed{\frac{13}{28}}$. The closure property holds.

d) Sum: $\boxed{-\frac{10}{21}}$. The closure property holds.