Question

How do you solve the exponential inequality `5^(2x+2)>=25^(2x-3)`?

Answer 1

`x <=4`

Explanation:

Note that `25=5^2`
So `25^(2x-3)=(5^2)^(2x-3)=5^(4x-6)`

Therefore
`color(white)("XXX")5^(2x+2) >= 25^(2x-3)`
is equivalent to
`color(white)("XXX")5^(2x+2) >=5^(4x-6)`

which will be true if
`color(white)("XXX")2x+2 >= 4x-6`

Subtracting `2x` from both sides then adding `6` (to both sides)
`color(white)("XXX")8 >= 2x`

or
`color(white)("XXX")x <=4`