document.write( "Question 1209650: Determine the set of all real x satisfying
\n" ); document.write( "x^3 - 2x^2 - 3x < -25x^2 + 17x.
\n" ); document.write( "Enter your answer in interval notation.
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Algebra.Com's Answer #849742 by CPhill(1959)\"\" \"About 
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1. **Rewrite the inequality:**
\n" ); document.write( " Move all terms to one side:
\n" ); document.write( " x³ - 2x² - 3x + 25x² - 17x < 0
\n" ); document.write( " x³ + 23x² - 20x < 0\r
\n" ); document.write( "\n" ); document.write( "2. **Factor out x:**
\n" ); document.write( " x(x² + 23x - 20) < 0\r
\n" ); document.write( "\n" ); document.write( "3. **Solve the quadratic equation:**
\n" ); document.write( " Find the roots of x² + 23x - 20 = 0 using the quadratic formula:
\n" ); document.write( " x = (-b ± √(b² - 4ac)) / 2a
\n" ); document.write( " x = (-23 ± √(23² - 4(1)(-20))) / 2(1)
\n" ); document.write( " x = (-23 ± √(529 + 80)) / 2
\n" ); document.write( " x = (-23 ± √609) / 2
\n" ); document.write( " The roots are approximately x ≈ 0.839 and x ≈ -23.839\r
\n" ); document.write( "\n" ); document.write( "4. **Determine the intervals:**
\n" ); document.write( " 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:
\n" ); document.write( " * x < -23.839
\n" ); document.write( " * -23.839 < x < 0
\n" ); document.write( " * 0 < x < 0.839
\n" ); document.write( " * x > 0.839\r
\n" ); document.write( "\n" ); document.write( "5. **Test points in each interval:**
\n" ); document.write( " * x = -24: (-24)(-24² + 23(-24) - 20) < 0 => -24(576 - 552 - 20) < 0 => -24(4) < 0. True
\n" ); document.write( " * x = -1: (-1)(1 - 23 - 20) < 0 => (-1)(-42) < 0. False
\n" ); document.write( " * x = 0.5: (0.5)(0.25 + 11.5 - 20) < 0 => 0.5(-8.25) < 0. True
\n" ); document.write( " * x = 1: (1)(1 + 23 - 20) < 0 => 4 < 0. False\r
\n" ); document.write( "\n" ); document.write( "6. **Write the solution in interval notation:**
\n" ); document.write( " The inequality is satisfied when x < -23.839 or 0 < x < 0.839.\r
\n" ); document.write( "\n" ); document.write( "Final Answer: The final answer is $\boxed{(-\infty, \frac{-23 - \sqrt{609}}{2}) \cup (0, \frac{-23 + \sqrt{609}}{2})}$
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