Solve the inequality and express the answer in interval notation:
$-24 \leq 2x - 10 < -6$
The goal is to find the set of all real numbers \(x\) that satisfy the compound inequality \(-24 \leq 2x - 10 < -6\). To do this, we need to isolate the variable \(x\) in the middle of the inequality by performing the same operations on all three parts. This compound inequality represents two conditions that must both be true: \(2x - 10 \geq -24\) and \(2x - 10 < -6\). Solving these simultaneously allows us to find the range of possible values for \(x\).
To isolate the term with \(x\), we first add 10 to all parts of the inequality:
\[-24 + 10 \leq 2x - 10 + 10 < -6 + 10\]
Simplifying this yields:
\[-14 \leq 2x < 4\]
Now, we divide all three parts of the inequality by 2 to solve for \(x\):
\[\frac{-14}{2} \leq \frac{2x}{2} < \frac{4}{2}\]
This results in:
\[-7 \leq x < 2\]
This means that \(x\) can be any value from \(-7\) (inclusive) up to, but not including, 2.
To express this result in interval notation, we use a square bracket '[' for the inclusive end and a parenthesis '(' for the exclusive end. Therefore, the set of all real numbers \(x\) satisfying the inequality is \([-7, 2)\). The final answer is \(\boxed{[-7, 2)}\).
In real-life scenarios, interval notation is used to define ranges for various quantities. For example, a quality control process for a machine part might specify that the thickness of a steel plate must be between 2 mm and 4 mm to be acceptable. If the required thickness \(T\) must satisfy \(2 \leq T < 4\), this is represented as the interval \([2, 4)\). Using interval notation helps engineers, manufacturers, and data analysts clearly define boundaries for measurements, production tolerances, or acceptable parameter ranges in scientific experiments.