Solve the compound inequality:
$4 x-3>9$ or $2 x+1 \leq-5$
We are given a compound inequality involving the logical operator "or", which means we need to find all real numbers \(x\) that satisfy at least one of the two given inequalities: \(4x - 3 > 9\) or \(2x + 1 \leq -5\). The goal is to solve each inequality independently and then combine their solution sets into a single union of intervals.
The first inequality is \(4x - 3 > 9\). To isolate \(x\), we first add 3 to both sides:
\[4x > 9 + 3\]
\[4x > 12\]
Next, divide both sides by 4:
\[x > \frac{12}{4}\]
\[x > 3\]
So, the solution set for the first inequality is all numbers greater than 3, which in interval notation is \((3, \infty)\).
The second inequality is \(2x + 1 \leq -5\). To isolate \(x\), we first subtract 1 from both sides:
\[2x \leq -5 - 1\]
\[2x \leq -6\]
Next, divide both sides by 2:
\[x \leq \frac{-6}{2}\]
\[x \leq -3\]
So, the solution set for the second inequality is all numbers less than or equal to -3, which in interval notation is \((-\infty, -3]\).
The original statement is a disjunction, meaning we take the union of the two solution sets obtained. The union of the sets \((-\infty, -3]\) and \((3, \infty)\) is written as:
\[(-\infty, -3] \cup (3, \infty)\]
This represents all real numbers \(x\) such that \(x \leq -3\) or \(x > 3\).
Imagine you are testing the safety limit of a mechanical component in a machine. You determine that the component will fail if the operating temperature \(x\) is in a range considered "critical." Through testing, you find the machine is unstable if the temperature is 3 degrees or colder, or if the temperature exceeds 3 degrees Celsius above its reference point. Solving this compound inequality helps technicians define the "safe" operating temperature range, ensuring the machine functions correctly and safely.