Solve (x+1)(x-2)>(x-5)(x+5)

$$( x + 1 ) ( x - 2 ) > ( x - 5 ) ( x + 5 )$$

Solve for x

$$x<23$$

Solution Steps

Use the distributive property to multiply x+1 by x-2 and combine like terms.

$$x^{2}-x-2>\left(x-5\right)\left(x+5\right)$$

Consider \left(x-5\right)\left(x+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 5.

$$x^{2}-x-2>x^{2}-25$$

Subtract x^{2} from both sides.

$$x^{2}-x-2-x^{2}>-25$$

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

$$-x-2>-25$$

Add 2 to both sides.

$$-x>-25+2$$

Add -25 and 2 to get -23.

$$-x>-23$$

Divide both sides by -1. Since -1 is negative, the inequality direction is changed.

$$x<\frac{-23}{-1}$$

Fraction \frac{-23}{-1} can be simplified to 23 by removing the negative sign from both the numerator and the denominator.

$$x<23$$