**Quote:**

Can't you divide both quantities by (*x* + 2) ... ?

No, we cannot, because

we do NOT know if it is positive, negative or even 0!

You cannot divide by 0. If (

*x* + 2) = 0, then the quantities are equal.

If (

*x* + 2) < 0, then by dividing the both parts of the equation by a negative number, you change the sign to an opposite one. Since 5 > 4, then the original sign must be 5(

*x* + 2) < 4(

*x* + 2).

If (

*x* + 2) > 0, then by dividing the both parts of the equation by a positive number, you do NOT change the sign. Since 5 > 4, then the original sign must be 5(

*x* + 2) > 4(

*x* + 2).

You may try to plug in some values of

*x* less than -2 and greater than -2 to get the full understanding.

IF there had been 5(

*x*² + 1) and 4(

*x*² + 1), then we could have divided the inequality by (

*x*² + 1). (

*x*² + 1) is always positive. The sign would NOT have changed.

5(

*x*² + 1) ? 4(

*x*² + 1)

5 > 4

So 5(

*x*² + 1) > 4(

*x*² + 1)

IF there had been 5(-

*x*² – 1) and 4(-

*x*² – 1), then we either could have factored (-1) first, or we could have divided the inequality by (-

*x*² – 1) right away. (-

*x*² – 1) is always negative, so the sign would have changed.

5(-

*x*² – 1) ? 4(-

*x*² – 1)

5 > 4

So 5(-

*x*² – 1) < 4(-

*x*² – 1)