Exponential inequalities and base direction — 9709 P3
Use increasing or decreasing behaviour before choosing an inequality direction.
Notes and key methods
The exponential \(\displaystyle a^{x}\) is increasing when a>1 and decreasing when 0<a<1. Comparing powers in the same base therefore preserves or reverses the inequality, respectively.
After comparing exponents, solve the linear inequality, taking account of its coefficient sign as well. A two-sided level restriction can be checked by ordering the resulting input endpoints and testing an interior point.
Check the conditions
The base is positive and not equal to one.
The coefficient p is positive in these exercises.
Three worked examples
Example 1 · Exponential inequalities
Let \(\displaystyle a=\)\(2\), \(\displaystyle p=2, h=5, k=6, q=1\). Solve \(a^{p(x-h)}>a^k\).
Solve the exponential inequality.
Which is the solution of the stated strict inequality?
\(x = 8\)
All real x
\(x < 8\)
\(x > 8\)
Find the boundary input for the level \(\displaystyle a^{k+q}\).
Find the width of the input interval between the two level boundaries.
Show the reasoning
(a)
Compare p(x−h) with k, reversing the direction exactly when 0<a<1; then divide by positive p.
Answer: \(x > 8\)
(b)
Use monotonicity of the exponential.
The base increases with its exponent; compare exponents accordingly.