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Exponential growth is a process where something increases by a constant percentage over equal time periods. Many natural phenomena, including population growth, follow this pattern. In our problem, both China and India's populations are growing exponentially, but at different rates. To describe exponential growth mathematically, we use functions. For China's population, you start with 1.34 billion and multiply it by an annual growth factor of 1.004. Similarly for India, the starting population is 1.19 billion, and the annual growth factor is 1.0137.
These functions allow us to predict how populations will grow in the future. They look like this:

• China: \( P_c(t) = 1.34 \times (1.004)^t \)
• India: \( P_i(t) = 1.19 \times (1.0137)^t \)

The t in these equations represents the number of years after 2011. Thus, understanding exponential growth is key to predicting when India's population will eventually surpass that of China.

To find out when India's population will exceed China's, we used inequality solving. An inequality is a mathematical statement that shows one expression is greater than or less than another. In this case, we're interested in when India's population expression becomes greater than China's. This is expressed with the inequality:
\[ 1.19 \times (1.0137)^t > 1.34 \times (1.004)^t \]
The process involves manipulating both sides of this inequality to isolate the growth factors on one side. By simplifying, dividing, and rearranging terms, we end up with:
\[ (1.0137/1.004)^t > 1.34/1.19 \]
This form of an inequality lets us directly compare the relative growth speeds of the two populations, crucial for determining when India overtakes China.

Algebraic manipulation involves changing the structure of an equation or inequality to make it easier to solve. In this exercise, it was necessary to break down the initial inequality to a form where we could isolate the variable t. This was done by dividing both sides by certain constants and rearranging equations.
Next, we utilized logarithms, which are inverse operations to exponents, to handle the exponential parts. Taking natural logarithms on both sides transforms the exponential inequality into a simple linear one:
\[ t \cdot \ln(1.0095118) > \ln(1.12605) \]
The process involves isolating t by dividing both sides by \( \ln(1.0095118) \). This manipulation allows us to solve for t directly, leading to:
\[ t > \frac{\ln(1.12605)}{\ln(1.0095118)} \]
A key takeaway is that algebraic manipulation is crucial when dealing with exponential inequalities, as regular algebraic methods won't suffice.