Solve the inequality:
$-2x - 4 < -16$
(There may be more than one correct answer)
We are given the linear inequality \(-2x - 4 < -16\). Our goal is to determine the range of values for \(x\) that satisfy this inequality. A linear inequality is a mathematical statement where one expression is less than or greater than another. The process of solving it is similar to solving linear equations, but with the crucial rule that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign.
To isolate the term involving \(x\), we first add 4 to both sides of the inequality. This operation maintains the inequality's relationship:
\[-2x - 4 + 4 < -16 + 4\]
After simplifying both sides, we get:
\[-2x < -12\]
Now, to solve for \(x\), we divide both sides by -2. Since we are dividing by a negative number, we must reverse the direction of the inequality sign from \(<\) to \(>\):
\[\frac{-2x}{-2} > \frac{-12}{-2}\]
Performing the division gives:
\[x > 6\]
This means that any real number \(x\) greater than 6 satisfies the original inequality.
Finally, we can verify our solution. For instance, if we pick a number greater than 6, such as \(x = 7\), the original inequality becomes:
\[-2(7) - 4 < -16\]
\[-14 - 4 < -16\]
\[-18 < -16\]
Since \(-18\) is indeed less than \(-16\), our solution \(x > 6\) is correct. The set of all values is \(\boxed{x > 6}\).
Linear inequalities like this one are common in scenarios where you have a "limit" or a "budget." For example, imagine you are managing a small monthly budget for office supplies, where you start with a reserve and spend \(\$2\) for each unit of a supply you buy. If you know that your reserve balance minus the cost of units (\(-2x - 4\)) must be less than \(\$-16\) for the company to trigger a special maintenance alert, you can use this inequality to determine how many units (\(x\)) you need to purchase to trigger that alert. Knowing that \(x > 6\) means you need to order more than 6 units to hit that specific financial state.