1. **State the problem:** Solve the inequality $$\frac{2x}{x+1} < -2$$ using the Case Analysis Method. 2. **Rewrite the inequality in the form $$\frac{P(x)}{Q(x)} < 0$$:** Start by moving all terms to the left side: $$\frac{2x}{x+1} + 2 < 0$$ 3. **Combine into a single rational expression:** Find a common denominator $x+1$: $$\frac{2x}{x+1} + \frac{2(x+1)}{x+1} < 0$$ Simplify numerator: $$\frac{2x + 2(x+1)}{x+1} < 0$$ $$\frac{2x + 2x + 2}{x+1} < 0$$ $$\frac{4x + 2}{x+1} < 0$$ So, $$P(x) = 4x + 2$$ and $$Q(x) = x + 1$$. 4. **Find critical points:** Set numerator and denominator equal to zero: $$4x + 2 = 0 \Rightarrow x = -\frac{1}{2}$$ $$x + 1 = 0 \Rightarrow x = -1$$ These points divide the number line into intervals: $$(-\infty, -1), (-1, -\frac{1}{2}), (-\frac{1}{2}, \infty)$$ 5. **Analyze the sign of $$\frac{4x+2}{x+1}$$ on each interval:** - For $$x < -1$$, pick $$x = -2$$: $$P(-2) = 4(-2) + 2 = -8 + 2 = -6 < 0$$ $$Q(-2) = -2 + 1 = -1 < 0$$ Fraction sign: $$\frac{negative}{negative} = positive$$ (not less than zero) - For $$-1 < x < -\frac{1}{2}$$, pick $$x = -0.75$$: $$P(-0.75) = 4(-0.75) + 2 = -3 + 2 = -1 < 0$$ $$Q(-0.75) = -0.75 + 1 = 0.25 > 0$$ Fraction sign: $$\frac{negative}{positive} = negative$$ (less than zero, solution interval) - For $$x > -\frac{1}{2}$$, pick $$x = 0$$: $$P(0) = 2 > 0$$ $$Q(0) = 1 > 0$$ Fraction sign: $$\frac{positive}{positive} = positive$$ (not less than zero) 6. **Check points where denominator is zero:** At $$x = -1$$, denominator zero, so $$x = -1$$ is excluded. At $$x = -\frac{1}{2}$$, numerator zero, fraction equals zero, but inequality is strict $$< 0$$, so exclude $$x = -\frac{1}{2}$$. 7. **Write the solution in set builder notation:** $$\{ x \mid -1 < x < -\frac{1}{2} \}$$ **Final answer:** $$\boxed{\{ x \mid -1 < x < -\frac{1}{2} \}}$$