Solve the inequality and express the answer in interval notation:
-31 < 9x + 5 < 32
The given problem requires solving the compound linear inequality \(-31 < 9x + 5 < 32\). A compound inequality of this form means that the expression \(9x + 5\) must satisfy two conditions simultaneously: \(9x + 5 > -31\) and \(9x + 5 < 32\). Our primary goal is to isolate the variable \(x\) in the middle of the inequality. To do this, we will apply algebraic operations to all parts of the inequality consistently, ensuring the inequality signs remain in the correct direction.
First, we isolate the term containing \(x\) by removing the constant \(5\) from the middle. We do this by subtracting \(5\) from all three parts of the inequality:
\[-31 - 5 < 9x + 5 - 5 < 32 - 5\]
Performing the subtraction, we obtain:
\[-36 < 9x < 27\]
Now that we have the term \(9x\) isolated, we need to solve for \(x\) by dividing all parts of the inequality by the coefficient \(9\). Since \(9\) is a positive number, the direction of the inequality signs will remain unchanged:
\[\frac{-36}{9} < \frac{9x}{9} < \frac{27}{9}\]
Calculating these divisions gives us:
\[-4 < x < 3\]
The solution to the inequality is all real numbers \(x\) such that \(x\) is greater than \(-4\) and less than \(3\). In interval notation, this set of values is expressed as the open interval \((-4, 3)\). Thus, the solution is \(\boxed{(-4, 3)}\).
Compound inequalities are frequently used in engineering and manufacturing to define tolerance ranges for parts. For example, if a machine part is designed to be \(9x+5\) millimeters long, where \(x\) represents a specific calibration factor, the engineering specifications might require that the length be strictly between \(31\) mm below and \(32\) mm above a baseline setting to ensure the part fits correctly. By solving the compound inequality, engineers determine the acceptable range of calibration values for \(x\), ensuring the final product meets safety and performance standards.