Answer:
3x > 3x^2 + 6x + 2
Next, let's simplify the equation by moving all terms to one side:
0 > 3x^2 + 6x + 2 - 3x
0 > 3x^2 + 3x + 2
Now, we have a quadratic inequality. To solve it, we can follow these steps:
1. Factor the quadratic equation if possible. In this case, the quadratic equation doesn't factor nicely, so we move to the next step.
2. Find the critical points by setting the quadratic equation equal to zero and solving for x:
3x^2 + 3x + 2 = 0
Using the quadratic formula, we can find the solutions:
x = (-3 ± √(3^2 - 4(3)(2))) / (2(3))
Simplifying the equation gives us:
x = (-3 ± √(-15)) / 6
Since we have a negative value under the square root, the solutions will be complex numbers. However, the question asks for the values of x, so we can continue with the next step.
3. Analyze the sign of the quadratic equation. We can do this by plotting the quadratic function or using a sign chart. Alternatively, we can examine the coefficient of x^2, which is positive (3). This tells us that the parabola opens upward.
4. Determine the regions where the quadratic is positive or negative. Since the parabola opens upward and we have a greater than (>) sign in the inequality, we are interested in the regions where the quadratic is positive.
5. Identify the values of x in the positive regions. To find the x-values where y = 3x exceeds y = 3x^2 + 6x + 2, we need to determine where the quadratic is negative. This means we need to find the intervals where the quadratic is positive and take the complement of those intervals.
Based on the analysis, we can conclude that the values of x where y = 3x exceeds y = 3x^2 + 6x + 2 are the complement of the intervals where the quadratic is positive.
Explanation: