Let's solve each equation step by step.

### a) \( x + \frac{1}{9} = -\frac{7}{16} + \frac{1}{9} \)

First, combine the fractions on the right-hand side:

\[ -\frac{7}{16} + \frac{1}{9} \]

To combine these, find a common denominator. The least common multiple of 16 and 9 is 144.

\[ -\frac{7}{16} = -\frac{7 \times 9}{16 \times 9} = -\frac{63}{144} \]
\[ \frac{1}{9} = \frac{1 \times 16}{9 \times 16} = \frac{16}{144} \]

Now combine them:

\[ -\frac{63}{144} + \frac{16}{144} = -\frac{47}{144} \]

So the equation becomes:

\[ x + \frac{1}{9} = -\frac{47}{144} \]

Subtract \(\frac{1}{9}\) from both sides:

\[ x = -\frac{47}{144} - \frac{1}{9} \]

Convert \(\frac{1}{9}\) to a fraction with a denominator of 144:

\[ \frac{1}{9} = \frac{16}{144} \]

So:

\[ x = -\frac{47}{144} - \frac{16}{144} = -\frac{63}{144} \]

Simplify:

\[ x = -\frac{63}{144} = -\frac{7}{16} \]

### b) \( \frac{1}{2}x - \frac{3}{5} = -\frac{1}{5} \)

Add \(\frac{3}{5}\) to both sides:

\[ \frac{1}{2}x = -\frac{1}{5} + \frac{3}{5} \]

Combine the fractions on the right-hand side:

\[ -\frac{1}{5} + \frac{3}{5} = \frac{2}{5} \]

So:

\[ \frac{1}{2}x = \frac{2}{5} \]

Multiply both sides by 2 to solve for \( x \):

\[ x = 2 \times \frac{2}{5} = \frac{4}{5} \]

### c) \( \frac{31}{2} - \frac{1}{2}x = \frac{2}{3} \)

Subtract \(\frac{31}{2}\) from both sides:

\[ -\frac{1}{2}x = \frac{2}{3} - \frac{31}{2} \]

Convert \(\frac{31}{2}\) to a fraction with a denominator of 6:

\[ \frac{31}{2} = \frac{31 \times 3}{2 \times 3} = \frac{93}{6} \]

So:

\[ -\frac{1}{2}x = \frac{2}{3} - \frac{93}{6} \]

Convert \(\frac{2}{3}\) to a fraction with a denominator of 6:

\[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \]

So:

\[ -\frac{1}{2}x = \frac{4}{6} - \frac{93}{6} = -\frac{89}{6} \]

Multiply both sides by -2 to solve for \( x \):

\[ x = -2 \times -\frac{89}{6} = \frac{178}{6} = \frac{89}{3} \]

### d) \( \frac{5}{9}x - \frac{1}{3}x = \frac{1}{8} + \frac{11}{9}x \)

Combine the \( x \)-terms on the left-hand side:

\[ \frac{5}{9}x - \frac{1}{3}x = \frac{5}{9}x - \frac{3}{9}x = \frac{2}{9}x \]

So the equation becomes:

\[ \frac{2}{9}x = \frac{1}{8} + \frac{11}{9}x \]

Subtract \(\frac{11}{9}x\) from both sides:

\[ \frac{2}{9}x - \frac{11}{9}x = \frac{1}{8} \]

Combine the \( x \)-terms:

\[ -\frac{9}{9}x = \frac{1}{8} \]

Simplify:

\[ -x = \frac{1}{8} \]

Multiply both sides by -1 to solve for \( x \):

\[ x = -\frac{1}{8} \]

### e) \( x - \frac{7}{3} + x - \frac{7}{4} = x - \frac{7}{5} + x - \frac{7}{6} \)

Combine like terms on both sides:

\[ 2x - \left( \frac{7}{3} + \frac{7}{4} \right) = 2x - \left( \frac{7}{5} + \frac{7}{6} \right) \]

Simplify the constants:

\[ \frac{7}{3} + \frac{7}{4} = \frac{28}{12} + \frac{21}{12} = \frac{49}{12} \]
\[ \frac{7}{5} + \frac{7}{6} = \frac{42}{30} + \frac{35}{30} = \frac{77}{30} \]

So the equation becomes:

\[ 2x - \frac{49}{12} = 2x - \frac{77}{30} \]

Since the \( 2x \) terms cancel out, we are left with:

\[ -\frac{49}{12} = -\frac{77}{30} \]

This is not true, so there is no solution to this equation.