- Rewrite the bases \(1/4\) and \(1/8\) as powers of 2: \(2^{-2}\) and \(2^{-3}\).
- Simplify the inequality to \(-2m + 8 > -3m + 48\), which yields the condition \(m > 40\).
- Test each value in the set \(S = \{-32, -8, 16, 32\}\) and find that none are greater than 40.
- Conclude that none of the given integers make the original inequality false. \(\boxed{\text{None}}\)
Explanation
- Problem Analysis and Negation
The original inequality is given by \(\left(\frac{1}{4}\right)^{m-4} \le \left(\frac{1}{8}\right)^{m-16}\). We want to identify which integers \(m\) from the set \(S = \{-32, -8, 16, 32\}\) make this statement false. An inequality is false when its negation is true, so we are looking for values of \(m \in S\) that satisfy:
\[\left(\frac{1}{4}\right)^{m-4} > \left(\frac{1}{8}\right)^{m-16}\]
- Simplifying the Exponential Expression
To solve the inequality \(\left(\frac{1}{4}\right)^{m-4} > \left(\frac{1}{8}\right)^{m-16}\), we rewrite both sides with a common base of 2:
\[\left(2^{-2}\right)^{m-4} > \left(2^{-3}\right)^{m-16}\]
Using the power of a power rule \((a^b)^c = a^{bc}\), we expand the exponents:
\[2^{-2(m-4)} > 2^{-3(m-16)}\]
\[2^{-2m+8} > 2^{-3m+48}\]
- Solving the Inequality for m
Since the base \(2\) is greater than 1, the inequality \(2^A > 2^B\) is equivalent to \(A > B\). Therefore, we compare the exponents:
\[-2m + 8 > -3m + 48\]
Adding \(3m\) to both sides:
\[m + 8 > 48\]
Subtracting 8 from both sides:
\[m > 40\]
The original inequality is false if and only if \(m > 40\).
- Conclusion and Evaluation of Options
We check each element \(m \in S = \{-32, -8, 16, 32\}\) against the condition \(m > 40\):
- For \(m = -32\): \(-32 > 40\) is false.
- For \(m = -8\): \(-8 > 40\) is false.
- For \(m = 16\): \(16 > 40\) is false.
- For \(m = 32\): \(32 > 40\) is false.
Upon reviewing the calculations, it appears that none of the integers in the provided set \(S = \{-32, -8, 16, 32\}\) satisfy the condition \(m > 40\). Thus, none of the options provided make the inequality false. However, re-evaluating the initial inequality \(\left(\frac{1}{4}\right)^{m-4} \le \left(\frac{1}{8}\right)^{m-16}\), we see that if none satisfy the negation, then all satisfy the original inequality. Let us re-verify: if \(m=32\), \((1/4)^{28} \approx 1.8 \times 10^{-17}\) and \((1/8)^{16} \approx 3.5 \times 10^{-15}\). Since \(1.8 \times 10^{-17} < 3.5 \times 10^{-15}\), the original inequality holds. Given the standard structure of such problems and the set provided, it is possible there is a typo in the inequality or the set provided in the prompt. Following the strictly calculated derivation, no values from the set make the statement false.
Examples
Inequalities like this appear when comparing exponential growth or decay rates, such as in finance (compound interest) or radioactive decay. For example, if you compare the decay rates of two different isotopes, you can determine the time \(t\) at which one isotope's concentration falls below the other's. Solving such inequalities helps in identifying thresholds for safety, efficiency, or resource management.