Solve x(x^2-4)-(x-2)(x^2+2x+4)-5/4>0

$$x ( x ^ { 2 } - 4 ) - ( x - 2 ) ( x ^ { 2 } + 2 x + 4 ) - \frac { 5 } { 4 } > 0$$

Solve for x

$$x<\frac{27}{16}$$

Solution Steps

Use the distributive property to multiply x by x^{2}-4.

$$x^{3}-4x-\left(x-2\right)\left(x^{2}+2x+4\right)-\frac{5}{4}>0$$

Use the distributive property to multiply x-2 by x^{2}+2x+4 and combine like terms.

$$x^{3}-4x-\left(x^{3}-8\right)-\frac{5}{4}>0$$

To find the opposite of x^{3}-8, find the opposite of each term.

$$x^{3}-4x-x^{3}+8-\frac{5}{4}>0$$

Combine x^{3} and -x^{3} to get 0.

$$-4x+8-\frac{5}{4}>0$$

Subtract \frac{5}{4} from 8 to get \frac{27}{4}.

$$-4x+\frac{27}{4}>0$$

Subtract \frac{27}{4} from both sides. Anything subtracted from zero gives its negation.

$$-4x>-\frac{27}{4}$$

Divide both sides by -4. Since -4 is negative, the inequality direction is changed.

$$x<\frac{-\frac{27}{4}}{-4}$$

Express \frac{-\frac{27}{4}}{-4} as a single fraction.

$$x<\frac{-27}{4\left(-4\right)}$$

Multiply 4 and -4 to get -16.

$$x<\frac{-27}{-16}$$

Fraction \frac{-27}{-16} can be simplified to \frac{27}{16} by removing the negative sign from both the numerator and the denominator.

$$x<\frac{27}{16}$$