1. **State the problem:** Solve the inequality $$\frac{2x - 3}{x + 5} \leq 0$$. 2. **Recall the rule for rational inequalities:** A rational expression $$\frac{A}{B} \leq 0$$ is less than or equal to zero where the numerator and denominator have opposite signs or the numerator is zero. 3. **Find critical points:** Set numerator and denominator equal to zero separately. - Numerator zero: $$2x - 3 = 0 \implies x = \frac{3}{2}$$ - Denominator zero: $$x + 5 = 0 \implies x = -5$$ (vertical asymptote, excluded from domain) 4. **Determine sign intervals:** The critical points divide the number line into three intervals: - $$(-\infty, -5)$$ - $$(-5, \frac{3}{2})$$ - $$(\frac{3}{2}, \infty)$$ 5. **Test each interval:** - For $$x < -5$$, pick $$x = -6$$: $$\frac{2(-6) - 3}{-6 + 5} = \frac{-12 - 3}{-1} = \frac{-15}{-1} = 15 > 0$$ (positive) - For $$-5 < x < \frac{3}{2}$$, pick $$x = 0$$: $$\frac{2(0) - 3}{0 + 5} = \frac{-3}{5} = -0.6 < 0$$ (negative) - For $$x > \frac{3}{2}$$, pick $$x = 2$$: $$\frac{2(2) - 3}{2 + 5} = \frac{4 - 3}{7} = \frac{1}{7} > 0$$ (positive) 6. **Include points where numerator is zero:** At $$x = \frac{3}{2}$$, the expression equals zero, so include this point. 7. **Exclude points where denominator is zero:** At $$x = -5$$, the expression is undefined, so exclude this point. 8. **Write the solution:** The inequality holds where the expression is less than or equal to zero, so $$ x \in \left(-5, \frac{3}{2}\right] $$ **Final answer:** $$\boxed{\left(-5, \frac{3}{2}\right]}$$