Solve 4x(x-6)geq0

$$4 x ( x - 6 ) \geq 0$$

Solve for x

$$x\in (-\infty,0]\cup [6,\infty)$$

Solution Steps

Use the distributive property to multiply 4x by x-6.

$$4x^{2}-24x\geq 0$$

Factor out x.

$$4x\left(x-6\right)\geq 0$$

For the product to be ≥0, x and x-6 have to be both ≤0 or both ≥0. Consider the case when x and x-6 are both ≤0.

$$x\leq 0 x-6\leq 0$$

The solution satisfying both inequalities is x\leq 0.

$$x\leq 0$$

Consider the case when x and x-6 are both ≥0.

$$x-6\geq 0 x\geq 0$$

The solution satisfying both inequalities is x\geq 6.

$$x\geq 6$$

The final solution is the union of the obtained solutions.

$$x\leq 0\text{; }x\geq 6$$