Solve the inequality:
$x+8<3 \text { or }-8 x<-40$
We are given a compound inequality involving the logical operator "or," which means we need to find the values of \(x\) that satisfy at least one of the two given inequalities: \(x + 8 < 3\) or \(-8x < -40\). The final solution set will be the union of the solution sets of these two individual inequalities.
We start with the first inequality: \(x + 8 < 3\). To solve for \(x\), we subtract 8 from both sides of the inequality:
\[x + 8 - 8 < 3 - 8\]
\[x < -5\]
Thus, the first part of our solution set is the interval \((-\infty, -5)\).
Next, we solve the second inequality: \(-8x < -40\). To isolate \(x\), we divide both sides by -8. A crucial rule in algebra is that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. Thus, the less-than sign (\(<\)) becomes a greater-than sign (\(>\)):
\[x > \frac{-40}{-8}\]
\[x > 5\]
Thus, the second part of our solution set is the interval \((5, \infty)\).
The complete solution for the compound inequality is the union of the two individual solution sets found in the previous steps. Since we are looking for values of \(x\) that satisfy either \(x < -5\) or \(x > 5\), the final solution is the union of the sets \((-\infty, -5)\) and \((5, \infty)\). In interval notation, this is expressed as \((-\infty, -5) \cup (5, \infty)\).
Consider a scenario where a machine is calibrated to operate within a specific temperature range, but it must avoid a "danger zone." If the safe operating temperatures are below \(50^\circ C\) or above \(80^\circ C\), we can model the unsafe temperatures using an inequality. If the machine's temperature \(T\) must satisfy \(T < 50\) or \(T > 80\), this is analogous to our inequality \(x < -5\) or \(x > 5\). Solving such inequalities helps technicians ensure that equipment stays within safe operational parameters, preventing overheating or failure.