1. **State the problem:** Solve the inequality $$x^1 - 4x^3 - 7x^2 + 22x + 24 < 0$$. 2. **Rewrite the inequality in standard polynomial form:** $$-4x^3 - 7x^2 + 23x + 24 < 0$$ 3. **Find the roots of the cubic polynomial** $$-4x^3 - 7x^2 + 23x + 24 = 0$$ to determine intervals where the polynomial changes sign. 4. **Try rational root theorem candidates:** Possible roots are factors of 24 divided by factors of 4, i.e., $$\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \pm24, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{3}{4}, \pm\frac{1}{4}$$. 5. **Test $x=2$:** $$-4(2)^3 -7(2)^2 + 23(2) + 24 = -4(8) -7(4) + 46 + 24 = -32 -28 + 46 + 24 = 10 \neq 0$$ 6. **Test $x=-2$:** $$-4(-2)^3 -7(-2)^2 + 23(-2) + 24 = -4(-8) -7(4) -46 + 24 = 32 -28 -46 + 24 = -18 \neq 0$$ 7. **Test $x=3$:** $$-4(3)^3 -7(3)^2 + 23(3) + 24 = -4(27) -7(9) + 69 + 24 = -108 -63 + 69 + 24 = -78 \neq 0$$ 8. **Test $x=-1$:** $$-4(-1)^3 -7(-1)^2 + 23(-1) + 24 = -4(-1) -7(1) -23 + 24 = 4 -7 -23 + 24 = -2 \neq 0$$ 9. **Test $x=4$:** $$-4(4)^3 -7(4)^2 + 23(4) + 24 = -4(64) -7(16) + 92 + 24 = -256 -112 + 92 + 24 = -252 \neq 0$$ 10. **Test $x=-3$:** $$-4(-3)^3 -7(-3)^2 + 23(-3) + 24 = -4(-27) -7(9) -69 + 24 = 108 -63 -69 + 24 = 0$$ So, $x=-3$ is a root. 11. **Divide the polynomial by $(x+3)$:** Using polynomial division or synthetic division: $$\frac{-4x^3 -7x^2 + 23x + 24}{x+3} = -4x^2 + 5x + 8$$ 12. **Solve the quadratic equation:** $$-4x^2 + 5x + 8 = 0$$ Multiply both sides by $-1$ for easier handling: $$4x^2 - 5x - 8 = 0$$ 13. **Use quadratic formula:** $$x = \frac{5 \pm \sqrt{(-5)^2 - 4(4)(-8)}}{2 \times 4} = \frac{5 \pm \sqrt{25 + 128}}{8} = \frac{5 \pm \sqrt{153}}{8}$$ 14. **Approximate roots:** $$\sqrt{153} \approx 12.37$$ So, $$x_1 = \frac{5 - 12.37}{8} = \frac{-7.37}{8} \approx -0.92$$ $$x_2 = \frac{5 + 12.37}{8} = \frac{17.37}{8} \approx 2.17$$ 15. **Roots of the cubic are:** $$x = -3, -0.92, 2.17$$ 16. **Determine sign intervals:** Test values in intervals: - For $x < -3$, test $x = -4$: $$-4(-4)^3 -7(-4)^2 + 23(-4) + 24 = -4(-64) -7(16) -92 + 24 = 256 -112 -92 + 24 = 76 > 0$$ - For $-3 < x < -0.92$, test $x = -2$: $$-4(-2)^3 -7(-2)^2 + 23(-2) + 24 = 32 -28 -46 + 24 = -18 < 0$$ - For $-0.92 < x < 2.17$, test $x = 0$: $$-4(0) -7(0) + 0 + 24 = 24 > 0$$ - For $x > 2.17$, test $x = 3$: $$-108 -63 + 69 + 24 = -78 < 0$$ 17. **Solution to inequality $< 0$ is where polynomial is negative:** $$(-3, -0.92) \cup (2.17, \infty)$$ **Final answer:** $$\boxed{(-3, \frac{5 - \sqrt{153}}{8}) \cup (\frac{5 + \sqrt{153}}{8}, \infty)}$$