Solve the inequality. Write the solution set in interval notation if possible. Simplify your answer, if necessary. \[ 8-9 x>4 x^{2} \] The solution set is \( \square \). (ロ, \( \square \) [ \( [\square, \square][\square, \square) \)

To solve the inequality \( 8 - 9x > 4x^2 \), first rearrange it into standard form: \[ 4x^2 + 9x - 8 < 0. \] Next, we can factor this quadratic inequality. The appropriate factors will lead us to identify the roots of the equation \( 4x^2 + 9x - 8 = 0 \). Using the quadratic formula, where \( a = 4, b = 9, \text{ and } c = -8 \): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 4 \cdot (-8)}}{2 \cdot 4}. \] \[ = \frac{-9 \pm \sqrt{81 + 128}}{8} = \frac{-9 \pm \sqrt{209}}{8}. \] Calculating the approximate values gives us the two roots: \[ x_1 \approx -0.427, \quad x_2 \approx -4.573. \] Now we can test values in the intervals \( (-\infty, -4.573) \), \( (-4.573, -0.427) \), and \( (-0.427, \infty) \) to check where the expression is negative. Through testing, we find that the solution set where \( 4x^2 + 9x - 8 < 0 \) is: \[ (-4.573, -0.427). \] The solution set is \( (-4.573, -0.427). \) And just like that, we have our interval! 🌈 Keep in mind that it’s crucial to confirm which intervals give a negative result to ensure our final answer is correct. Happy solving!