Question

What is the solution to x⁴+4x³ <12x²?

A) (−6,2)
B) [−6,2]
C) (−[infinity],−6)∪(2,[infinity])
D) (−[infinity],−6]∪[2,[infinity])

Answer 1

To solve the inequality x⁴ + 4x³ < 12x², we rearrange and factor it to find the intervals where the inequality is satisfied, leading to the solution C) (−∞, −6) ∪ (2, ∞). The correct option is C

Step-by-step explanation:

The solution to the inequality x⁴ + 4x³ < 12x² involves rearranging the inequality and finding the values of x that satisfy it. First, we can subtract 12x² from both sides of the inequality to obtain x⁴ + 4x³ - 12x² < 0. Factoring the left-hand side, we get x²(x² + 4x - 12) < 0. Factoring further, we find x²(x + 6)(x - 2) < 0. The roots of the equation x²(x + 6)(x - 2) = 0 are x = 0, x = -6, and x = 2. By testing intervals determined by these roots, we can determine where the inequality is satisfied. The solution to the inequality is the union of intervals where the expression is negative, which in this case is (−∞, −6) ∪ (2, ∞). Therefore, the correct answer is C) (−∞, −6) ∪ (2, ∞).