- Solve the first inequality \(-1 - 10a < -1\) to get \(a > 0\).
- Solve the second inequality \(10 + 3a \leq -5\) to get \(a \leq -5\).
- Combine these results using the "or" condition to find the solution set: \(a > 0\) or \(a \leq -5\).
- The final answer is \(\boxed{a > 0 \text{ or } a \leq -5}\).
Explanation
- Analysis of the Compound Inequality
The problem asks us to solve the compound inequality \(-1 - 10a < -1\) or \(10 + 3a \leq -5\). A compound inequality with "or" means that the solution set is the union of the solutions to each individual inequality. We will solve each inequality separately and then combine the results.
- Solving the First Inequality
First, we solve \(-1 - 10a < -1\). Adding \(1\) to both sides of the inequality gives \(-10a < 0\). To isolate \(a\), we divide both sides by \(-10\). Since we are dividing by a negative number, we must reverse the inequality sign. Thus, we get \(a > 0\).
- Solving the Second Inequality
Next, we solve \(10 + 3a \leq -5\). Subtracting \(10\) from both sides gives \(3a \leq -15\). Dividing both sides by \(3\) yields \(a \leq -5\).
- Combining and Selecting the Correct Answer
The solution to the compound inequality is the union of the two results: \(a > 0\) or \(a \leq -5\). Comparing this result to the given options, we find that it matches the fourth option: \(a > 0\) or \(a \leq -5\).
Examples
Compound inequalities are often used in engineering and safety standards. For instance, imagine a machine that operates safely only if the internal temperature \(T\) is either very cold (below a certain threshold) or hot (above a certain safety range, like during a high-temperature cycle). If the safety constraint is \(T < -10^\circ C\) or \(T > 50^\circ C\), this is a perfect example of a compound inequality defining safe operating regions. Calculating these ranges ensures equipment reliability and operator safety.