Express the following inequality in interval notation:
$-24 \leq 2 x-10<-6$
We are given the compound inequality \(-24 \leq 2x - 10 < -6\). Our objective is to find the set of all real numbers \(x\) that satisfy this condition and express the solution in interval notation. A compound inequality like this represents two conditions that must be satisfied simultaneously: \(-24 \leq 2x - 10\) and \(2x - 10 < -6\). We will solve this by isolating \(x\) in the middle part of the inequality.
To isolate \(2x\), we first add 10 to every part of the inequality:
\[-24 + 10 \leq 2x - 10 + 10 < -6 + 10\]
Simplifying each part, we get:
\[-14 \leq 2x < 4\]
Next, to solve for \(x\), we divide all parts of the inequality by 2:
\[\frac{-14}{2} \leq \frac{2x}{2} < \frac{4}{2}\]
This simplifies to:
\[-7 \leq x < 2\]
Now that we have the simplified inequality \(-7 \leq x < 2\), we can write the solution set in interval notation. The inequality \(-7 \leq x\) means that \(-7\) is included in the set, which we represent with a square bracket '['. The inequality \(x < 2\) means that 2 is not included, which we represent with a parenthesis ')'. Therefore, the solution set in interval notation is \([-7, 2)\).
Interval notation is frequently used in fields like physics and engineering to define ranges of operation. For example, if a machine is designed to operate safely within a temperature range of \([-7^\circ\text{C}, 2^\circ\text{C})\), it means the machine is safe at \(-7^\circ\text{C}\) but must be kept strictly below \(2^\circ\text{C}\) to prevent damage. Understanding this notation helps technicians define operational limits clearly and safely.