Solve -27xleq-12x^2

$$- 27 x \leq - 12 x ^ { 2 }$$

Solve for x

$$x\in \begin{bmatrix}0,\frac{9}{4}\end{bmatrix}$$

Solution Steps

Add 12x^{2} to both sides.

$$-27x+12x^{2}\leq 0$$

Factor out x.

$$3x\left(4x-9\right)\leq 0$$

For the product to be ≤0, one of the values x and x-\frac{9}{4} has to be ≥0 and the other has to be ≤0. Consider the case when x\geq 0 and x-\frac{9}{4}\leq 0.

$$x\geq 0 x-\frac{9}{4}\leq 0$$

The solution satisfying both inequalities is x\in \left[0,\frac{9}{4}\right].

$$x\in \begin{bmatrix}0,\frac{9}{4}\end{bmatrix}$$

Consider the case when x\leq 0 and x-\frac{9}{4}\geq 0.

$$x-\frac{9}{4}\geq 0 x\leq 0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}0,\frac{9}{4}\end{bmatrix}$$