Final answer:
To find the solution to the quadratic inequality 6x²≥5−13x, use the quadratic formula to find the solutions of the equation 6x²-13x+5=0, and determine the values of x that satisfy the inequality. The solution to the quadratic inequality is (-[infinity], -5/2] ∪ [13/2, [infinity]), which is option D.
Step-by-step explanation:
To find the solution to the quadratic inequality 6x²≥5-13x, we can follow these steps:
- Multiply both sides of the inequality by -1 to make the coefficient of x² positive: -6x²≤-5+13x
- Rearrange the terms to get a quadratic inequality in standard form: 6x²-13x+5≤0
- Factorize the quadratic if possible. In this case, the quadratic cannot be factored easily.
- Apply the quadratic formula to find the solutions of the equation 6x²-13x+5=0: x = (-b±√(b²-4ac))/2a
- Determine the value(s) of x that satisfy the inequality. Since the inequality is ≤0, we are looking for the values of x that make the quadratic expression less than or equal to zero.
Using the quadratic formula, we find that the solutions of the equation are x = (13±√(104))/12. Plugging these values into the inequality, we find that the solution to the quadratic inequality is (-[infinity], -5/2] ∪ [13/2, [infinity]). Therefore, the correct answer is option D. (-[infinity], -5/2] ∪ [13/2, [infinity]).