If x is a negative integer, which of the following expressions has the LEAST value?

Question: If x is a negative integer, which of the following expressions has the LEAST value?

Solution:
Given that x is a negative integer, so x < 0 and |x| ≥ 1.

Now we compare each expression:
First, consider x² - 0.5. Since x is a non-zero integer, x² ≥ 1. Therefore, x² - 0.5 ≥ 0.5, which is always positive.
Next, consider x² - 1.5. Since x² ≥ 1, we get x² - 1.5 ≥ - 0.5.
So this expression is never very small (it is always greater than or equal to - 0.5).

Now consider x. Since x is a negative integer, it is less than 0.
Now consider x³ - 1.5. For any negative integer x, x³ is also negative and in fact x³ is always less than x.

Therefore:
x³ - 1.5 < x³ < x
So x³ - 1.5 is smaller than x.

Also, we already know both x² - 0.5 and x² - 1.5 are greater than or equal to - 0.5, while x³ - 1.5 is less than x, which is negative.
Hence x³ - 1.5 is smaller than all the other expressions.

Therefore, the least value is x³ - 1.5.