Rational Inequality 577395

1. **State the problem:** Solve the inequality $$\frac{2x - 5}{x + 1} \geq 3$$. 2. **Rewrite the inequality:** Move all terms to one side to compare to zero: $$\frac{2x - 5}{x + 1} - 3 \geq 0$$ 3. **Find a common denominator:** $$\frac{2x - 5}{x + 1} - \frac{3(x + 1)}{x + 1} \geq 0$$ 4. **Combine the fractions:** $$\frac{2x - 5 - 3(x + 1)}{x + 1} \geq 0$$ 5. **Simplify the numerator:** $$2x - 5 - 3x - 3 = -x - 8$$ So the inequality is: $$\frac{-x - 8}{x + 1} \geq 0$$ 6. **Rewrite numerator for clarity:** $$\frac{-(x + 8)}{x + 1} \geq 0$$ 7. **Analyze critical points:** The numerator is zero at $x = -8$, denominator zero at $x = -1$ (excluded from domain). 8. **Determine sign intervals:** - For $x < -8$, numerator $-(x+8) > 0$, denominator $x+1 < 0$, fraction $> 0$ or $< 0$? - For $-8 < x < -1$, numerator $-(x+8) < 0$, denominator $x+1 < 0$, fraction $> 0$ or $< 0$? - For $x > -1$, numerator $-(x+8) < 0$, denominator $x+1 > 0$, fraction $> 0$ or $< 0$? 9. **Test each interval:** - $x = -9$: numerator $-( -9 + 8) = -(-1) = 1 > 0$, denominator $-9 + 1 = -8 < 0$, fraction $\frac{+}{-} = - < 0$ - $x = -5$: numerator $-( -5 + 8) = -(3) = -3 < 0$, denominator $-5 + 1 = -4 < 0$, fraction $\frac{-}{-} = + > 0$ - $x = 0$: numerator $-(0 + 8) = -8 < 0$, denominator $0 + 1 = 1 > 0$, fraction $\frac{-}{+} = - < 0$ 10. **Include points where numerator is zero:** At $x = -8$, fraction equals zero, so included. 11. **Domain restriction:** $x \neq -1$ because denominator zero. 12. **Final solution:** $$[-8, -1)$$ **Answer:** The inequality holds for all $x$ in the interval $$[-8, -1)$$.