Final answer:
To solve the inequality x²(-4 9) ≤ -1/4 (36x + 24), we set up a quadratic equation and use the quadratic formula. However, since the discriminant is negative, there are no real solutions for x. Therefore, the solution set for x²(-4 9) ≤ -1/4 (36x + 24) is the empty set.
Step-by-step explanation:
To solve the inequality x²(-4 9) ≤ -1/4 (36x + 24), we can start by expanding the left side:
x²(-4 9) ≤ -1/4 (36x + 24)
-4x² + 9x² ≤ -9x - 6
Combining like terms, we have:
5x² ≤ -9x - 6
Next, we bring all the terms to one side to set up a quadratic equation:
5x² + 9x + 6 ≤ 0
Now we can use the quadratic formula to find the solutions for x:
x = (-b ± √(b² - 4ac))/(2a)
In this case, a = 5, b = 9, and c = 6. Plugging these values into the quadratic formula, we get two solutions:
x = (-9 ± √(9² - 4(5)(6)))/(2(5))
x = (-9 ± √(81 - 120))/(10)
x = (-9 ± √(-39))/(10)
Since the discriminant (-39) is negative, there are no real solutions for x. Therefore, the solution set for x²(-4 9) ≤ -1/4 (36x + 24) is the empty set.