2x²>=-4x

To solve the inequality 2x² >= -4x, we can start by simplifying the inequality as much as possible.

First, let's move all the terms to the left side of the inequality:

2x² + 4x >= 0

Next, we can factor out a common factor of 2x:

2x(x + 2) >= 0

Now, we have a product (2x) multiplied by another expression (x + 2) which is equal to or greater than zero.

To determine the solution, we need to consider three cases:

Case 1: When both factors are positive: 2x >= 0 and (x + 2) >= 0

For 2x >= 0, dividing both sides by 2 gives: x >= 0

For (x + 2) >= 0, subtracting 2 from both sides gives: x >= -2

So in this case, the solution is x >= -2.

Case 2: When both factors are negative: 2x <= 0 and (x + 2) <= 0

For 2x <= 0, dividing both sides by 2 gives: x <= 0

For (x + 2) <= 0, subtracting 2 from both sides gives: x <= -2

So in this case, there is no solution.

Case 3: When one factor is positive and the other is negative: 2x <= 0 and (x + 2) >= 0

For 2x <= 0, dividing both sides by 2 gives: x <= 0

For (x + 2) >= 0, subtracting 2 from both sides gives: x >= -2

So in this case, the solution is -2 <= x <= 0.

Combining all three cases, the solution to the inequality 2x² >= -4x is -2 <= x <= 0.