Rational numbers part I

1. Subtract $(-11)/5$ from additive inverse of sum of $-5/8$ and $5/6$.

Firstly, calculate the sum of $-5/8$ and $5/6$:

$$ -\frac{5}{8} + \frac{5}{6} = -\frac{15}{24} + \frac{20}{24} = \frac{5}{24} $$

The additive inverse of this sum is $-\frac{5}{24}$. Now, subtract $(-11)/5$ from $-\frac{5}{24}$:

$$ -\frac{5}{24} - (-\frac{11}{5}) = -\frac{5}{24} + \frac{11}{5} = -\frac{25}{120} + \frac{264}{120} = \frac{239}{120} = \frac{119}{60} $$

2. Subtract the sum of $(-4)/7$ and $5/14$ from the sum of $9/14$ and $23/28$.

Firstly, calculate the sum of $(-4)/7$ and $5/14$:

$$ -\frac{4}{7} + \frac{5}{14} = -\frac{8}{14} + \frac{5}{14} = -\frac{3}{14} $$

Next, calculate the sum of $9/14$ and $23/28$:

$$ \frac{9}{14} + \frac{23}{28} = \frac{18}{28} + \frac{23}{28} = \frac{41}{28} $$

Now, subtract $-\frac{3}{14}$ from $\frac{41}{28}$:

$$ \frac{41}{28} - (-\frac{3}{14}) = \frac{41}{28} + \frac{6}{28} = \frac{47}{28} = \frac{47}{2} $$

3. What number should be added to $-7/8$ so as to get $5/9$?

To find the number that should be added to $-7/8$ to get $5/9$, simply subtract $-7/8$ from $5/9$:

$$ \frac{5}{9} - \left(-\frac{7}{8}\right) = \frac{5}{9} + \frac{7}{8} = \frac{40}{72} + \frac{63}{72} = \frac{103}{72} $$

4. Verify the distributive law of multiplication over subtraction for the rational numbers x=$(-3)/4$, y=$5/7$, z=$(-4)/5$.

The distributive law of multiplication over subtraction states that for any numbers x, y, and z, we have $x \cdot (y - z) = x \cdot y - x \cdot z$.

Therefore, substituting the given values, we get:

$$ -\frac{3}{4} \cdot \left(\frac{5}{7} - -\frac{4}{5}\right) = -\frac{3}{4} \cdot \frac{5}{7} - -\frac{3}{4} \cdot -\frac{4}{5} $$

Now, compute each side:

$$ -\frac{3}{4} \cdot \left(\frac{5}{7} + \frac{4}{5}\right) = -\frac{3}{4} \cdot \left(\frac{25}{35} + \frac{28}{35}\right) = -\frac{3}{4} \cdot \frac{53}{35} = -\frac{159}{140} $$ $$ -\frac{3}{4} \cdot \frac{5}{7} - -\frac{3}{4} \cdot -\frac{4}{5} = -\frac{15}{28} - \frac{12}{20} = -\frac{159}{140} $$

Since the left-hand side equals the right-hand side, this verifies the distributive law of multiplication over subtraction for the given rational numbers.

5. Write 6 rational numbers between $3/(-7)$ and $2/5$.

We can find 6 rational numbers between $3/(-7)$ and $2/5$ as: $-6/35$, $-4/35$, $-2/35$, $1/35$, $3/35$, $5/35$.