# Exponential Inequality

How to solve an exponential inequality: 2 examples and their solutions.

## Example2^{5x - 9} > 4^{x}

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.

## Example(1/16)⋅(1/8)^{x} ≤ (1/4)^{x}

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.