Inequality in Exponential Growth?

📌 CONCEPT: Exponential growth is a type of growth where the rate of growth is proportional to the current quantity, often modeled using linear inequalities to determine the range of values for certain variables.

📐 RULE / FORMULA: The rule for exponential growth can be represented by the inequality P(t) > P0 ∗ (1 + r)^t, where P(t) is the population at time t, P0 is the initial population, r is the growth rate, and t is the time in years.

💡 WORKED EXAMPLE: Given an initial population of 500 individuals growing at a rate of 20% per year, we can use the inequality P(t) > 1000 to find the range of years in which the population will exceed 1000. Substituting the given values, we get 500 ∗ (1 + 0.20)^t > 1000. Simplifying, we get (1.20)^t > 2. This can be further simplified using logarithms to find the range of t.

⚠️ COMMON MISTAKE: Students often forget to check the direction of the inequality when solving exponential inequalities, leading to incorrect calculations and conclusions.