Solve (x-6)(x-9)-(x-7)(x-1)<7(2x-5)

$$( x - 6 ) ( x - 9 ) - ( x - 7 ) ( x - 1 ) < 7 ( 2 x - 5 )$$

Solve for x

$$x>\frac{82}{21}$$

Solution Steps

Use the distributive property to multiply x-6 by x-9 and combine like terms.

$$x^{2}-15x+54-\left(x-7\right)\left(x-1\right)<7\left(2x-5\right)$$

Use the distributive property to multiply x-7 by x-1 and combine like terms.

$$x^{2}-15x+54-\left(x^{2}-8x+7\right)<7\left(2x-5\right)$$

To find the opposite of x^{2}-8x+7, find the opposite of each term.

$$x^{2}-15x+54-x^{2}+8x-7<7\left(2x-5\right)$$

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

$$-15x+54+8x-7<7\left(2x-5\right)$$

Combine -15x and 8x to get -7x.

$$-7x+54-7<7\left(2x-5\right)$$

Subtract 7 from 54 to get 47.

$$-7x+47<7\left(2x-5\right)$$

Use the distributive property to multiply 7 by 2x-5.

$$-7x+47<14x-35$$

Subtract 14x from both sides.

$$-7x+47-14x<-35$$

Combine -7x and -14x to get -21x.

$$-21x+47<-35$$

Subtract 47 from both sides.

$$-21x<-35-47$$

Subtract 47 from -35 to get -82.

$$-21x<-82$$

Divide both sides by -21. Since -21 is negative, the inequality direction is changed.

$$x>\frac{-82}{-21}$$

Fraction \frac{-82}{-21} can be simplified to \frac{82}{21} by removing the negative sign from both the numerator and the denominator.

$$x>\frac{82}{21}$$