Solve the inequality:
$-6 \leq 2 - 2x < 10$
To solve the compound inequality \(-6 \leq 2 - 2x < 10\), we need to isolate the variable \(x\). A compound inequality like this is a shorthand way of representing two separate inequalities simultaneously: \(-6 \leq 2 - 2x\) and \(2 - 2x < 10\). Our goal is to manipulate the inequality to find the range of values for \(x\).
First, we subtract 2 from all parts of the inequality to begin isolating the term with \(x\):
\[-6 - 2 \leq 2 - 2x - 2 < 10 - 2\]
This simplifies to:
\[-8 \leq -2x < 8\]
Next, we need to solve for \(x\) by dividing by \(-2\). It is a fundamental property of inequalities that dividing or multiplying by a negative number reverses the direction of the inequality signs. Therefore:
\[\frac{-8}{-2} \geq \frac{-2x}{-2} > \frac{8}{-2}\]
Performing the divisions gives:
\[4 \geq x > -4\]
To present the answer in standard form, we write it from the smallest value to the largest value:
\[-4 < x \leq 4\]
In interval notation, this set of values for \(x\) is written as \((-4, 4]\). The parenthesis \('('\) indicates that \(-4\) is not included, while the square bracket \(']'\) indicates that \(4\) is included.
This type of compound inequality is useful in various real-life scenarios, such as temperature control. For example, if a sensitive electronic component must operate within a safe temperature range, and the internal temperature is controlled by an automated system modeled by \(T = 2 - 2x\) (where \(x\) might represent a voltage adjustment), you would use this math to determine the necessary voltage input range to keep the component between -6 and 10 degrees. Knowing the safe range \((-4, 4]\) allows engineers to calibrate the system effectively to prevent overheating or freezing.