- Define the function \(h(x) = 10^x + 0.6^x - 7\), so the inequality becomes \(h(x) \leq 0\).
- Evaluate \(h(x)\) for the given options: \(h(-4) \approx 0.716\), \(h(-3) \approx -2.369\), \(h(0) = -5\), and \(h(1) = 3.6\).
- Identify that \(h(x) \leq 0\) for \(x = -3\) and \(x = 0\).
- Conclude that the least integer satisfying the inequality is \(-3\), so the answer is \(\boxed{-3}\).
Explanation
- Problem Analysis and Definition
To find the least integer value in the solution set of the inequality \(10^x \leq -0.6^x + 7\), we can define a function \(h(x) = 10^x + 0.6^x - 7\). The given inequality is equivalent to solving \(h(x) \leq 0\). We need to test the provided candidate integers (a) \(-4\), (b) \(-3\), (c) \(0\), and (d) \(1\) to see which ones satisfy the condition \(h(x) \leq 0\).
- Testing Integer Values
We evaluate the function \(h(x) = 10^x + 0.6^x - 7\) at each of the given integer values:
- For \(x = -4\): \(h(-4) = 10^{-4} + 0.6^{-4} - 7 = \frac{1}{10000} + \frac{1}{0.6^4} - 7 \approx 0.0001 + 7.716 - 7 = 0.7161\). Since \(0.7161 > 0\), the value \(x = -4\) does not satisfy the inequality.
- For \(x = -3\): \(h(-3) = 10^{-3} + 0.6^{-3} - 7 = \frac{1}{1000} + \frac{1}{0.216} - 7 \approx 0.001 + 4.630 - 7 = -2.369\). Since \(-2.369 \leq 0\), the value \(x = -3\) satisfies the inequality.
- For \(x = 0\): \(h(0) = 10^0 + 0.6^0 - 7 = 1 + 1 - 7 = -5\). Since \(-5 \leq 0\), the value \(x = 0\) satisfies the inequality.
- For \(x = 1\): \(h(1) = 10^1 + 0.6^1 - 7 = 10 + 0.6 - 7 = 3.6\). Since \(3.6 > 0\), the value \(x = 1\) does not satisfy the inequality.
- Identifying the Least Integer Value
Based on the evaluations, the integers \(-3\) and \(0\) satisfy the inequality \(10^x \leq -0.6^x + 7\). Comparing the values that satisfy the condition, we are looking for the least integer value among these, which is \(-3\). Thus, the smallest integer value in the solution set from the given options is \(-3\).
Examples
This problem helps us understand how to solve inequalities involving exponential functions, which are critical in fields like finance for calculating interest, biology for modeling population growth, and physics for analyzing decay processes. By finding the boundary of the inequality, we can determine the range of conditions where a specific growth model is constrained by a limit, similar to finding the maximum capacity a server can handle or the minimum time until a reaction reaches completion. Mastering these algebraic techniques provides a foundation for analyzing complex systems and decision-making.