Question

X^3>x^2 and |x| > 1, then

1. X > 1

2. X < -1

3. -1 < X < 1

4. X < -1 or X > 1

5. X < -1 and X > 1

Answer 1

1. X > 1

Explanation:

We know
`x^2 > 1` because
`x^2 = |x|^2 > 1`. That's why

`X^3 > x^2 > 1`.

We can now subtract 1 in both sides of the inequality:

`X^3 -1 > 0`.

Factoring
`X^3 -1` as a difference of cubes, we get:

`(X-1)(X^2 + X + 1) > 0`.

Thus, we have two factors whose multiplication is positive. Then, both are positive or both are negative. The second case is impossible, because
`X^2 + X + 1` can never be positive. The reason is the following:

`X^2 + X + 1 = 2 (X^2 + X + 1)/(2) = (2X^2 + 2X + 2)/(2) = (X^2 + 2X + 1 +X^2 + 1)/(2) = ((X+1)^2 +X^2 + 1)/(2)`

witch is always positive because
`(X+1)^2 + X^2 + 1` is a sum of squares, and the squares are always positive.

We conclude that both
`(X-1)` and
`(X^2 + X + 1)` have to be positive, and then
`X-1 > 0` implies
`X > 1`