Solve 3x^2-xgeqx^2-5x

$$3 x ^ { 2 } - x \geq x ^ { 2 } - 5 x$$

Solve for x

$$x\in (-\infty,-2]\cup [0,\infty)$$

Solution Steps

Subtract x^{2} from both sides.

$$3x^{2}-x-x^{2}\geq -5x$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$2x^{2}-x\geq -5x$$

Add 5x to both sides.

$$2x^{2}-x+5x\geq 0$$

Combine -x and 5x to get 4x.

$$2x^{2}+4x\geq 0$$

Factor out x.

$$2x\left(x+2\right)\geq 0$$

For the product to be ≥0, x+2 and x have to be both ≤0 or both ≥0. Consider the case when x+2 and x are both ≤0.

$$x+2\leq 0 x\leq 0$$

The solution satisfying both inequalities is x\leq -2.

$$x\leq -2$$

Consider the case when x+2 and x are both ≥0.

$$x\geq 0 x+2\geq 0$$

The solution satisfying both inequalities is x\geq 0.

$$x\geq 0$$

The final solution is the union of the obtained solutions.

$$x\leq -2\text{; }x\geq 0$$