Solve the inequality below.
$-50<7 k+6<-8$
The given problem is to solve the compound linear inequality \(-50 < 7k + 6 < -8\). This inequality states that the expression \(7k + 6\) must be simultaneously greater than -50 and less than -8. To find the set of values for \(k\), we need to isolate the variable \(k\) within the expression by performing the same algebraic operations on all three parts of the inequality.
To isolate the term with \(k\), we first subtract 6 from all parts of the inequality:
\[-50 - 6 < 7k + 6 - 6 < -8 - 6\]
Simplifying the constants, we get:
\[-56 < 7k < -14\]
Next, to solve for \(k\), we divide all three parts of the inequality by 7:
\[\frac{-56}{7} < \frac{7k}{7} < \frac{-14}{7}\]
Performing the division, we obtain the range for \(k\):
\[-8 < k < -2\]
Comparing our result \(-8 < k < -2\) with the given options, we identify that the third option matches our derived range. Thus, the solution to the inequality is the interval between -8 and -2. The final answer is \(\boxed{-8 < k < -2}\).
Imagine you are calibrating a temperature sensor that must operate within a specific range. Suppose the sensor's raw output, denoted by \(7k + 6\) (where \(k\) is a control parameter), must stay between -50 degrees and -8 degrees to ensure safety. By solving the inequality \(-50 < 7k + 6 < -8\), you find that you must adjust your control parameter \(k\) to be between -8 and -2 for the sensor to function correctly. This type of calculation is common in engineering to ensure system stability and safety.