Dealing with inequalities with exponents always remember there are some traps,
- 1. The trend is not similar for |x| > 1 and |x| < 1
2. For negative x, the powers alternately change their signs too.
While dealing with exponents of variables always remember the following scenarios. Also try to understand them. Then you can solve this kind of problems just by looking at it.
For any +ve number x,
- 1. If x < 1, then repeated multiplication of the number with itself results in smaller quantity. Hence x > x^2 > x^3 > x^4 > ...
2. If x >1, then repeated multiplication of the number with itself results in larger quantity. Hence x < x^2 < x^3 < x^4 < ...
However for any -ve number x the scenario is a bit complicated due to alternate sign change.
- 1. If x < 1, then repeated multiplication of the number with itself results in smaller quantity but even powers are positive. Hence x^2 > x^4 > ... > x > x^3 > x^5 > ...
2. If x >1, then repeated multiplication of the number with itself results in larger quantity but even powers are positive. Hence ... x^5 < x^3 < x < x^2 < x^4 ...