Solve -3n+3n^2-nleq0

$$- 3 n + 3 n ^ { 2 } - n \leq 0$$

Solve for n

$$n\in \begin{bmatrix}0,\frac{4}{3}\end{bmatrix}$$

Solution Steps

Combine -3n and -n to get -4n.

$$-4n+3n^{2}\leq 0$$

Factor out n.

$$n\left(3n-4\right)\leq 0$$

For the product to be ≤0, one of the values n and n-\frac{4}{3} has to be ≥0 and the other has to be ≤0. Consider the case when n\geq 0 and n-\frac{4}{3}\leq 0.

$$n\geq 0 n-\frac{4}{3}\leq 0$$

The solution satisfying both inequalities is n\in \left[0,\frac{4}{3}\right].

$$n\in \begin{bmatrix}0,\frac{4}{3}\end{bmatrix}$$

Consider the case when n\leq 0 and n-\frac{4}{3}\geq 0.

$$n-\frac{4}{3}\geq 0 n\leq 0$$

This is false for any n.

$$n\in \emptyset$$

The final solution is the union of the obtained solutions.

$$n\in \begin{bmatrix}0,\frac{4}{3}\end{bmatrix}$$