To solve the inequalities \( a - 18 + 29a < -38 \) and \( a - \frac{89}{5} \geq -4 \), we will deal with each inequality separately.
1. Solving the first inequality:
\[ a - 18 + 29a < -38 \] Combine like terms: \[ 30a - 18 < -38 \] Add 18 to both sides: \[ 30a < -38 + 18 \] \[ 30a < -20 \] Divide both sides by 30: \[ a < -\frac{20}{30} \] Simplify: \[ a < -\frac{2}{3} \]
2. Solving the second inequality:
\[ a - \frac{89}{5} \geq -4 \] Add \(\frac{89}{5}\) to both sides: \[ a \geq -4 + \frac{89}{5} \] Convert \(-4\) to a fraction with a denominator of 5: \[ -4 = -\frac{20}{5} \] Now combine the fractions: \[ a \geq -\frac{20}{5} + \frac{89}{5} \] \[ a \geq \frac{89 - 20}{5} \] \[ a \geq \frac{69}{5} \]
Combine the results:
We have two parts:
- \( a < -\frac{2}{3} \)
- \( a \geq \frac{69}{5} \)
Final answer:
The solution set is: \[ \boxed{a < -\frac{2}{3} \text{ or } a \geq \frac{69}{5}} \]