Solve xleft(x^2-9right)-left(x-2right)left(x^2+2x+4right)-5/4>0

$$x \left( { x }^{ 2 } -9 \right) - \left( x-2 \right) \left( { x }^{ 2 } +2x+4 \right) - \frac{ 5 }{ 4 } > 0$$

Solve for x

$$x<\frac{3}{4}$$

Solution Steps

Use the distributive property to multiply x by x^{2}-9.

$$x^{3}-9x-\left(x-2\right)\left(x^{2}+2x+4\right)-\frac{5}{4}>0$$

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

$$x^{3}-9x-\left(x^{3}-8\right)-\frac{5}{4}>0$$

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

$$x^{3}-9x-x^{3}+8-\frac{5}{4}>0$$

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

$$-9x+8-\frac{5}{4}>0$$

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

$$-9x+\frac{27}{4}>0$$

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

$$-9x>-\frac{27}{4}$$

Divide both sides by -9. Since -9 is negative, the inequality direction is changed.

$$x<\frac{-\frac{27}{4}}{-9}$$

Express \frac{-\frac{27}{4}}{-9} as a single fraction.

$$x<\frac{-27}{4\left(-9\right)}$$

Multiply 4 and -9 to get -36.

$$x<\frac{-27}{-36}$$

Reduce the fraction \frac{-27}{-36} to lowest terms by extracting and canceling out -9.

$$x<\frac{3}{4}$$