Solve 3xgeq52x^2

$$3 x \geq 52 x ^ { 2 }$$

Solve for x

$$x\in \begin{bmatrix}0,\frac{3}{52}\end{bmatrix}$$

Solution Steps

Subtract 52x^{2} from both sides.

$$3x-52x^{2}\geq 0$$

Multiply the inequality by -1 to make the coefficient of the highest power in 3x-52x^{2} positive. Since -1 is negative, the inequality direction is changed.

$$-3x+52x^{2}\leq 0$$

Factor out x.

$$x\left(52x-3\right)\leq 0$$

For the product to be ≤0, one of the values x and x-\frac{3}{52} has to be ≥0 and the other has to be ≤0. Consider the case when x\geq 0 and x-\frac{3}{52}\leq 0.

$$x\geq 0 x-\frac{3}{52}\leq 0$$

The solution satisfying both inequalities is x\in \left[0,\frac{3}{52}\right].

$$x\in \begin{bmatrix}0,\frac{3}{52}\end{bmatrix}$$

Consider the case when x\leq 0 and x-\frac{3}{52}\geq 0.

$$x-\frac{3}{52}\geq 0 x\leq 0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}0,\frac{3}{52}\end{bmatrix}$$