Solve -14(1/2x-1)(1/2x+1)-1/2(1-7x^2)>0

$$- 14 ( \frac { 1 } { 2 } x - 1 ) ( \frac { 1 } { 2 } x + 1 ) - \frac { 1 } { 2 } ( 1 - 7 x ^ { 2 } ) > 0$$

Evaluate (complex solution)

Solution Steps

Use the distributive property to multiply -14 by \frac{1}{2}x-1.

$$\left(-7x+14\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(1-7x^{2}\right)>0$$

Use the distributive property to multiply -7x+14 by \frac{1}{2}x+1 and combine like terms.

$$-\frac{7}{2}x^{2}+14-\frac{1}{2}\left(1-7x^{2}\right)>0$$

Use the distributive property to multiply -\frac{1}{2} by 1-7x^{2}.

$$-\frac{7}{2}x^{2}+14-\frac{1}{2}+\frac{7}{2}x^{2}>0$$

Subtract \frac{1}{2} from 14 to get \frac{27}{2}.

$$-\frac{7}{2}x^{2}+\frac{27}{2}+\frac{7}{2}x^{2}>0$$

Combine -\frac{7}{2}x^{2} and \frac{7}{2}x^{2} to get 0.

$$\frac{27}{2}>0$$

Compare \frac{27}{2} and 0.

$$\text{true}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply -14 by \frac{1}{2}x-1.

$$\left(-7x+14\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(1-7x^{2}\right)>0$$

Use the distributive property to multiply -7x+14 by \frac{1}{2}x+1 and combine like terms.

$$-\frac{7}{2}x^{2}+14-\frac{1}{2}\left(1-7x^{2}\right)>0$$

Use the distributive property to multiply -\frac{1}{2} by 1-7x^{2}.

$$-\frac{7}{2}x^{2}+14-\frac{1}{2}+\frac{7}{2}x^{2}>0$$

Subtract \frac{1}{2} from 14 to get \frac{27}{2}.

$$-\frac{7}{2}x^{2}+\frac{27}{2}+\frac{7}{2}x^{2}>0$$

Combine -\frac{7}{2}x^{2} and \frac{7}{2}x^{2} to get 0.

$$\frac{27}{2}>0$$

This is true for any x.

$$x\in \mathrm{R}$$