Solve 3x^2-9x^2+3geq0

$$3 { x }^{ 2 } -9 { x }^{ 2 } +3 \geq 0$$

Solve for x

$$x\in \begin{bmatrix}-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\end{bmatrix}$$

Solution Steps

Combine 3x^{2} and -9x^{2} to get -6x^{2}.

$$-6x^{2}+3\geq 0$$

Multiply the inequality by -1 to make the coefficient of the highest power in -6x^{2}+3 positive. Since -1 is negative, the inequality direction is changed.

$$6x^{2}-3\leq 0$$

Add \frac{1}{2} to both sides.

$$x^{2}\leq \frac{1}{2}$$

Calculate the square root of \frac{1}{2} and get \frac{\sqrt{2}}{2}. Rewrite \frac{1}{2} as \left(\frac{\sqrt{2}}{2}\right)^{2}.

$$x^{2}\leq \left(\frac{\sqrt{2}}{2}\right)^{2}$$

Inequality holds for |x|\leq \frac{\sqrt{2}}{2}.

$$|x|\leq \frac{\sqrt{2}}{2}$$

Rewrite |x|\leq \frac{\sqrt{2}}{2} as x\in \left[-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right].

$$x\in \begin{bmatrix}-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\end{bmatrix}$$