Exponential Inequalities
Grade 10 · Algebra
Core Principle
Solving exponential inequalities reduces to comparing exponents once both sides are written as powers of the same base. The key tool is the monotonicity of the exponential function $f(x) = a^x$.
Increasing Function: $a > 1$
When the base $a > 1$, the function $f(x) = a^x$ is strictly increasing.
A larger function value corresponds to a larger argument.
- $a^x > a^b \;\Longleftrightarrow\; x > b$
- $a^x < a^b \;\Longleftrightarrow\; x < b$
Inequality sign is preserved.
Decreasing Function: $0 < a < 1$
When $0 < a < 1$, the function $f(x) = a^x$ is strictly decreasing.
A larger function value corresponds to a smaller argument.
- $a^x > a^b \;\Longleftrightarrow\; x < b$
- $a^x < a^b \;\Longleftrightarrow\; x > b$
Inequality sign is reversed.
Interactive: Monotonicity of $y = a^x$
Drag the slider to change the base $a$ and observe how the function changes from increasing to decreasing.
Interactive visualization — available in the full lesson