- Rewrite the inequality \(3^{2x-1} < 27\) as \(3^{2x-1} < 3^3\) by expressing 27 as a power of 3.
- Use the property of exponential functions with base greater than 1 to equate the exponents: \(2x - 1 < 3\).
- Solve the linear inequality: \(2x < 4\), which simplifies to \(x < 2\).
- Conclude that the solution set is \(\boxed{(-\infty, 2)}\).
Explanation
- Analyzing the Inequality
We are given the exponential inequality \(3^{2x-1} < 27\). Our goal is to determine the solution set for \(x\). We are provided with a table of values for the function \(f(x) = 3^{2x-1}\):
| \(x\) | \(y\) \) 0 | \(\frac{1}{3}\) \) 1 | 3 \) 2 | 27 \) 3 | 243 \) |
The function \(f(x) = 3^{2x-1}\) is an exponential function. Since the base \(3\) is greater than 1, the function is strictly increasing. This means that as \(x\) increases, \(f(x)\) also increases.
- Solving the Inequality
To solve the inequality \(3^{2x-1} < 27\), we can express both sides as powers of the same base. We know that \(27 = 3^3\). Thus, the inequality can be rewritten as:
\[3^{2x-1} < 3^3\]
Since the base \(3 > 1\), the exponential function \(f(x) = 3^x\) is strictly increasing, which allows us to compare the exponents directly. Therefore, the inequality holds if and only if:
\[2x - 1 < 3\]
Now, we solve this linear inequality for \(x\):
\[2x < 3 + 1\]
\[2x < 4\]
\[x < 2\]
This indicates that all values of \(x\) strictly less than 2 satisfy the inequality. In interval notation, this solution set is \((-\infty, 2)\).
- Verification and Conclusion
We can verify our result using the table provided. The table shows that for \(x=1\), \(f(1) = 3\), which is less than 27. For \(x=2\), \(f(2) = 27\), which is equal to (not less than) 27. Since the function is strictly increasing, for any \(x > 2\), \(f(x) > 27\). For any \(x < 2\), \(f(x) < 27\). Thus, the solution set is indeed all real numbers strictly less than 2. The correct option is \((-\infty, 2)\).
Examples
Exponential inequalities appear frequently in population modeling. For instance, if a bacteria population grows according to the function \(P(t) = 3^{2t-1}\) where \(t\) represents time in hours, you can use this math to determine the time interval during which the population remains below a certain threshold. For example, to find when the population is less than 27 units, you solve the inequality \(3^{2t-1} < 27\), helping researchers predict safe time frames for experiments or identify when the growth reaches a critical level.