To solve the inequality
\[ -\frac{2}{3}x - 10 < \frac{1}{3}, \]
we will first isolate the term involving \( x \).
- Add 10 to both sides:
\[ -\frac{2}{3}x < \frac{1}{3} + 10. \]
- Convert 10 to a fraction with a denominator of 3:
\[ 10 = \frac{30}{3}, \] so \[ \frac{1}{3} + 10 = \frac{1}{3} + \frac{30}{3} = \frac{31}{3}. \]
Now the inequality reads:
\[ -\frac{2}{3}x < \frac{31}{3}. \]
- Multiply both sides by \( -\frac{3}{2} \) (this reverses the inequality):
\[ x > \frac{31}{3} \cdot -\frac{3}{2}. \]
Calculating the right side:
\[ \frac{31 \cdot -3}{3 \cdot 2} = \frac{-93}{6} = -\frac{31}{2}. \]
Thus, the inequality simplifies to:
\[ x > -\frac{31}{2}. \]
- Final solution:
The solution to the inequality is
\[ x > -\frac{31}{2}. \]
This can also be expressed as:
\[ x > -15.5. \]