- Express the inequality \(\left(\frac{1}{2}\right)^{2x+1} \leq 4\) with the common base \(2\): \((2^{-1})^{2x+1} \leq 2^2\).
- Simplify the exponents to get \(2^{-2x-1} \leq 2^2\).
- Compare the exponents to form the linear inequality \(-2x - 1 \leq 2\).
- Solve the inequality to find \(x \geq -\frac{3}{2}\), which is \(\boxed{\left[-\frac{3}{2}, \infty\right)}\).
Explanation
- Analyzing the inequality and identifying common bases
To solve the exponential inequality \(\left(\frac{1}{2}\right)^{2x+1} \leq 4\), we first identify the bases of the exponential expressions. We notice that both \(\frac{1}{2}\) and \(4\) can be expressed as powers of the same base, which is \(2\). Specifically, \(\frac{1}{2} = 2^{-1}\) and \(4 = 2^2\). This allows us to rewrite the given inequality with a common base of \(2\).
- Simplifying the inequality using properties of exponents
By substituting the common base into the inequality, we have \((2^{-1})^{2x+1} \leq 2^2\). Using the power of a power property, \((a^m)^n = a^{m \times n}\), the left side simplifies to:
\[2^{-1 \times (2x+1)} = 2^{-2x-1}.\]
Thus, the inequality becomes \(2^{-2x-1} \leq 2^2\).
- Solving the resulting linear inequality for \(x\)
Because the base \(2\) is greater than \(1\), the exponential function \(f(t) = 2^t\) is strictly increasing. This means that if \(2^u \leq 2^v\), then \(u \leq v\). Applying this to our inequality, we obtain a linear inequality:
\[-2x - 1 \leq 2.\]
To solve for \(x\), we first add \(1\) to both sides:
\[-2x \leq 3.\]
Next, we divide both sides by \(-2\). Remember that when dividing or multiplying an inequality by a negative number, the direction of the inequality sign must be reversed. Thus, we get:
\[x \geq -\frac{3}{2}.\]
- Stating the final solution set
The inequality \(x \geq -\frac{3}{2}\) represents all real numbers that are greater than or equal to \(-\frac{3}{2}\). In interval notation, this is written as \(\left[-\frac{3}{2}, \infty\right)\). This matches one of the provided options.
Examples
Exponential inequalities are frequently used in fields like finance and biology. For example, if an investment grows at a specific rate or a population of bacteria triples over time, you can model the growth using exponential functions. Solving an inequality like \(\left(\frac{1}{2}\right)^{2x+1} \leq 4\) helps determine the time duration \(x\) during which a certain threshold of growth or decay is not exceeded, allowing for better resource management and planning.