Solve {left(-6sqrt{6}xright)}^2-44left(6x^2-18right)>0

$${ \left(-6 \sqrt{ 6 } x \right) }^{ 2 } -44 \left( 6 { x }^{ 2 } -18 \right) > 0$$

Solve for x

$$x\in \left(-\frac{\sqrt{66}}{2},\frac{\sqrt{66}}{2}\right)$$

Solution Steps

Expand \left(-6\sqrt{6}x\right)^{2}.

$$\left(-6\right)^{2}\left(\sqrt{6}\right)^{2}x^{2}-44\left(6x^{2}-18\right)>0$$

Calculate -6 to the power of 2 and get 36.

$$36\left(\sqrt{6}\right)^{2}x^{2}-44\left(6x^{2}-18\right)>0$$

The square of \sqrt{6} is 6.

$$36\times 6x^{2}-44\left(6x^{2}-18\right)>0$$

Multiply 36 and 6 to get 216.

$$216x^{2}-44\left(6x^{2}-18\right)>0$$

Use the distributive property to multiply -44 by 6x^{2}-18.

$$216x^{2}-264x^{2}+792>0$$

Combine 216x^{2} and -264x^{2} to get -48x^{2}.

$$-48x^{2}+792>0$$

Multiply the inequality by -1 to make the coefficient of the highest power in -48x^{2}+792 positive. Since -1 is negative, the inequality direction is changed.

$$48x^{2}-792<0$$

Add \frac{33}{2} to both sides.

$$x^{2}<\frac{33}{2}$$

Calculate the square root of \frac{33}{2} and get \frac{\sqrt{66}}{2}. Rewrite \frac{33}{2} as \left(\frac{\sqrt{66}}{2}\right)^{2}.

$$x^{2}<\left(\frac{\sqrt{66}}{2}\right)^{2}$$

Inequality holds for |x|<\frac{\sqrt{66}}{2}.

$$|x|<\frac{\sqrt{66}}{2}$$

Rewrite |x|<\frac{\sqrt{66}}{2} as x\in \left(-\frac{\sqrt{66}}{2},\frac{\sqrt{66}}{2}\right).

$$x\in \left(-\frac{\sqrt{66}}{2},\frac{\sqrt{66}}{2}\right)$$