SOLUTION: Determine the set of all real x satisfying x^3 - 2x^2 - 3x < -25x^2 + 17x. Enter your answer in interval notation.

Question 1209650: Determine the set of all real x satisfying
x^3 - 2x^2 - 3x < -25x^2 + 17x.
Enter your answer in interval notation.

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
1. **Rewrite the inequality:**
Move all terms to one side:
x³ - 2x² - 3x + 25x² - 17x < 0
x³ + 23x² - 20x < 0
2. **Factor out x:**
x(x² + 23x - 20) < 0
3. **Solve the quadratic equation:**
Find the roots of x² + 23x - 20 = 0 using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
x = (-23 ± √(23² - 4(1)(-20))) / 2(1)
x = (-23 ± √(529 + 80)) / 2
x = (-23 ± √609) / 2
The roots are approximately x ≈ 0.839 and x ≈ -23.839
4. **Determine the intervals:**
The roots of the cubic equation are x = 0, x ≈ 0.839, and x ≈ -23.839. These roots divide the number line into four intervals:
* x < -23.839
* -23.839 < x < 0
* 0 < x < 0.839
* x > 0.839
5. **Test points in each interval:**
* x = -24: (-24)(-24² + 23(-24) - 20) < 0 => -24(576 - 552 - 20) < 0 => -24(4) < 0. True
* x = -1: (-1)(1 - 23 - 20) < 0 => (-1)(-42) < 0. False
* x = 0.5: (0.5)(0.25 + 11.5 - 20) < 0 => 0.5(-8.25) < 0. True
* x = 1: (1)(1 + 23 - 20) < 0 => 4 < 0. False
6. **Write the solution in interval notation:**
The inequality is satisfied when x < -23.839 or 0 < x < 0.839.
Final Answer: The final answer is $\boxed{(-\infty, \frac{-23 - \sqrt{609}}{2}) \cup (0, \frac{-23 + \sqrt{609}}{2})}$

RELATED QUESTIONS
Determine the set of all real x satisfying x^2 + 3x >= -2 - 5x^2 + 10x - 13.... (answered by josgarithmetic)

Determine the set of all real x satisfying (x^2 - 5x + 3)^2 < 9 + 6x + x^2 Enter your... (answered by CPhill,ikleyn)

Determine the range of the function $(x) = -4x^2-8x-6 + 3x^2 - 17x + 10. Enter your... (answered by CPhill)

Find the domain of f(x)= (2x + 3)/(2x^2 - 3x - 27). Leave your answer in interval... (answered by funmath)

Give the domain and range of h(x) = x^{2} + 9 a) What is the domain of h(x)? (answered by ewatrrr)

Find all values of x in the interval [0, 2π] that satisfy the inequality. (Enter... (answered by lwsshak3)

Find all real x where 2 \cdot \frac{x - 5}{x - 3} > \frac{2x - 5}{x + 2} + 20. Give... (answered by greenestamps)