Solve the compound inequality.
$x-5<-10 \text { or } 2 x-4>6$
The problem asks us to solve the compound inequality \(x - 5 < -10 \text{ or } 2x - 4 > 6\). A compound inequality with the "or" connective means that any value of \(x\) that satisfies at least one of the individual inequalities is a solution to the entire expression. We will solve each inequality separately and then determine their union.
To solve the first inequality \(x - 5 < -10\), we isolate \(x\) by adding 5 to both sides:
\[x - 5 + 5 < -10 + 5\]
\[x < -5\]
Thus, the first part of the solution set is \((-\infty, -5)\).
To solve the second inequality \(2x - 4 > 6\), we first add 4 to both sides:
\[2x > 6 + 4\]
\[2x > 10\]
Next, divide by 2:
\[x > \frac{10}{2}\]
\[x > 5\]
Thus, the second part of the solution set is \((5, \infty)\).
The solution to the compound inequality \(x - 5 < -10 \text{ or } 2x - 4 > 6\) is the union of the two individual solution sets found: \(x < -5\) or \(x > 5\). Comparing this result with the given choices, it matches the first option. Therefore, the correct answer is \(x < -5 \text{ or } x > 5\).
Compound inequalities are very useful in engineering and safety applications. For example, consider a safety system for a boiler where the pressure \(P\) must be maintained within a safe operating range. If the system shuts down if the pressure is too low (e.g., \(P < 10 \text{ psi}\)) or too high (e.g., \(P > 50 \text{ psi}\)), this represents a compound "or" condition. By solving these inequalities, engineers can determine the specific, unsafe operating regions for the equipment, ensuring that it remains within its designated safe, steady state.