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(
x² + 1) and 4(
x² + 1), then we could have divided the inequality by (
x² + 1). (
x² + 1) is always positive. The sign would NOT have changed.
5(
x² + 1) ? 4(
x² + 1)
5 > 4
So 5(
x² + 1) > 4(
x² + 1)
IF there had been 5(-
x² – 1) and 4(-
x² – 1), then we either could have factored (-1) first, or we could have divided the inequality by (-
x² – 1) right away. (-
x² – 1) is always negative, so the sign would have changed.
5(-
x² – 1) ? 4(-
x² – 1)
5 > 4
So 5(-
x² – 1) < 4(-
x² – 1)