Solve the inequality below.
$-50 < 7k + 6 < -8$
The goal is to solve the compound inequality \(-50 < 7k + 6 < -8\) for the variable \(k\). A compound inequality means that both parts, \(-50 < 7k + 6\) and \(7k + 6 < -8\), must be satisfied simultaneously. To isolate \(k\), we will perform algebraic operations on all three parts of the inequality simultaneously. We are allowed to add or subtract the same value from all parts, and multiply or divide all parts by the same positive constant without changing the direction of the inequality symbols.
Starting with the inequality:
\[-50 < 7k + 6 < -8\]
To isolate the term with \(k\), we subtract 6 from all parts:
\[-50 - 6 < 7k + 6 - 6 < -8 - 6\]
This simplifies to:
\[-56 < 7k < -14\]
Next, to solve for \(k\), we divide all parts by 7. Since 7 is a positive number, the inequality signs remain in their current orientation:
\[\frac{-56}{7} < \frac{7k}{7} < \frac{-14}{7}\]
Performing the divisions gives:
\[-8 < k < -2\]
Comparing our result \(-8 < k < -2\) with the given options:
The result matches the third option, \(-8 < k < -2\). Thus, the range for \(k\) is \(-8 < k < -2\).
Consider a sensor in a factory that needs to maintain its signal intensity within a specific range to function correctly. If the signal intensity is given by the formula \(S = 7k + 6\), where \(k\) is a control parameter, and the sensor requires the signal to be between -50 and -8, solving the inequality \(-50 < 7k + 6 < -8\) helps engineers determine the acceptable range of the parameter \(k\). By solving this, they find that \(k\) must be between -8 and -2 for the sensor to operate within safe limits.