How do you solve #6x^2+3x>= -5x-3+x^2#?

Let'e rearrange the inequality

We factorise

We can build the sign chart

Therefore,

To solve the inequality 6x^2 + 3x >= -5x - 3 + x^2, follow these steps:

1. Combine like terms on both sides of the inequality to simplify it: 6x^2 + x^2 + 3x + 5x + 3 >= 0 7x^2 + 8x + 3 >= 0

2. Factor the quadratic expression on the left-hand side: (7x + 3)(x + 1) >= 0

3. Determine the critical points by setting each factor equal to zero and solving for x: 7x + 3 = 0 --> x = -3/7 x + 1 = 0 --> x = -1

4. Create intervals on the number line using the critical points: Interval 1: x < -3/7 Interval 2: -3/7 < x < -1 Interval 3: x > -1

5. Test a value from each interval in the original inequality to determine the solution set: For Interval 1, let x = -1: 6(-1)^2 + 3(-1) >= -5(-1) - 3 + (-1)^2 6 - 3 >= 5 - 3 + 1 3 >= 3 (true)

   For Interval 2, let x = -1/2: 6(-1/2)^2 + 3(-1/2) >= -5(-1/2) - 3 + (-1/2)^2 6(1/4) - 3/2 >= 5/2 - 3 + 1/4 3/2 - 3/2 >= 3/2 - 9/4 0 >= -3/4 (true)

   For Interval 3, let x = 0: 6(0)^2 + 3(0) >= -5(0) - 3 + (0)^2 0 >= -3 (true)

Therefore, the solution to the inequality 6x^2 + 3x >= -5x - 3 + x^2 is x <= -3/7 or -1 <= x <= 0.