Exponential Inequality

To solve an exponential inequality,
make the bases of both sides the same
and compare the exponents.

It's similar to solving an exponential equation.

The base of the left side 2^{5x - 9} is 2.

So change the base of the right side to 2:
4^x = 2^2x.

To solve this equation,
first move the right side terms to the left side.

2^{5x - 9} > 2^2x

The bases of both sides are the same.

The base is greater than 1.
So the order of the inequality sign
doesn't change: > → >.

Then 5x - 9 > 2x.

Solve 5x - 9 > 2x.

Move -9 to the right side.
Move 2x to the left side.

Then 3x > 9.

Divide both sides by 3.

3 is plus.
So the order of the inequality sign
doesn't change.

Then x > 3.

So
x > 3
is the answer.

The bases of the numbers are 1/16, 1/8, and 1/4.
So change the bases of the numbers to 1/2.

1/16 = (1/2)⁴
(1/8)^x = [(1/2)³]^x = (1/2)^3x

(1/4)^x = [(1/2)²]^x = (1/2)^2x

Power of a Power

(1/2)⁴⋅(1/2)^3x = (1/2)^{4 + 3x}

Product of Powers

(1/2)^{4 + 3x} ≤ (1/2)^2x

The bases of both sides are the same.

The base is between 0 and 1.
So the order of the inequality sign
does change: ≤ → ≥.

Then 4 + 3x ≥ 2x.

Solve 4 + 3x ≥ 2x.

Move 4 to the right side.
Move 2x to the left side.

Then x ≥ -4.

So
x ≥ -4
is the answer.