Exponential Inequalities

Exponential Inequalities

How to solve exponential inequalities problems: examples and their solutions.

Example 1

Solve the given inequality. 2^(5x - 9) > 4^x

Make both side's bases the same.

25x - 9 > 22x

Power of a power

Compare the exponents.

The bases of both sides (2) are greater than 1.

Then the order of the inequality sign doesn't change.

So 5x - 9 > 2x.

Example 2

Solve the given equation. (1/4)*(1/8)^x <= (1/16)^x

Make both side's bases the same.

2-2 - 3x ≤ 2-4x

Negative exponent

Power of a power

Product of powers

Compare the exponents.

The bases both sides (2) are greater than 1.

Then the order of the inequality sign doesn't change.

So -2 - 3x ≤ -4x.

Example 2: Another Solution

Solve the given equation. (1/4)*(1/8)^x <= (1/16)^x

Let's solve the same example
by making the base to 1/2.

(1/2)2 + 3x ≤ (1/2)4x

Power of a power

Product of powers

Compare the exponents.

The bases both sides (1/2) are in between 0 and 1.

Then the order of the inequality sign changes.

So 2 + 3x ≥ 4x.