Solve 4n^2-4(14n^2-47)>0

$$4 n ^ { 2 } - 4 ( 14 n ^ { 2 } - 47 ) > 0$$

Solve for n

$$n\in \left(-\frac{\sqrt{611}}{13},\frac{\sqrt{611}}{13}\right)$$

Solution Steps

Use the distributive property to multiply -4 by 14n^{2}-47.

$$4n^{2}-56n^{2}+188>0$$

Combine 4n^{2} and -56n^{2} to get -52n^{2}.

$$-52n^{2}+188>0$$

Multiply the inequality by -1 to make the coefficient of the highest power in -52n^{2}+188 positive. Since -1 is negative, the inequality direction is changed.

$$52n^{2}-188<0$$

Add \frac{47}{13} to both sides.

$$n^{2}<\frac{47}{13}$$

Calculate the square root of \frac{47}{13} and get \frac{\sqrt{611}}{13}. Rewrite \frac{47}{13} as \left(\frac{\sqrt{611}}{13}\right)^{2}.

$$n^{2}<\left(\frac{\sqrt{611}}{13}\right)^{2}$$

Inequality holds for |n|<\frac{\sqrt{611}}{13}.

$$|n|<\frac{\sqrt{611}}{13}$$

Rewrite |n|<\frac{\sqrt{611}}{13} as n\in \left(-\frac{\sqrt{611}}{13},\frac{\sqrt{611}}{13}\right).

$$n\in \left(-\frac{\sqrt{611}}{13},\frac{\sqrt{611}}{13}\right)$$