Lesson 3 of 9, 10 min read
Exponential inequalities
Solving exponential inequalities by comparing exponents, keeping or reversing the inequality depending on the base, including quadratic-in-form inequalities.
The base decides the direction
Exponential functions are always increasing or always decreasing, so an inequality between two powers with the same base becomes an inequality between the exponents. But the direction depends on the base.
Always or never
Since \(b^x > 0\) for every \(x\):
- \(b^x > -3\) is true for all real numbers;
- \(b^x \le 0\) has no solution.
Quadratic in form
Practice
Quick check
What is the solution of \(\left(\frac{1}{3}\right)^x > 9\)?
\(9 = \left(\frac{1}{3}\right)^{-2}\), and the base is less than \(1\), so the sign reverses.
Quick check
For which \(x\) is \(4^x > 0\)?
A power with a positive base is always positive.
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