Which number line represents the solutions to $-2|x|=-6$?
The given equation is \(-2|x| = -6\). Our goal is to determine the values of \(x\) that satisfy this equation and visualize them on a number line. The absolute value function \(|x|\) represents the distance of a number \(x\) from zero on the number line, and it is always non-negative, meaning \(|x| \geq 0\). We need to isolate the absolute value expression first.
To isolate \(|x|\), we divide both sides of the equation by \(-2\):
\[\frac{-2|x|}{-2} = \frac{-6}{-2}\]
This simplifies to:
\[|x| = 3\]
The absolute value \(|x|\) equals 3, meaning the distance of \(x\) from zero on the number line is 3 units. This occurs when \(x\) is 3 units to the right of zero or 3 units to the left of zero. Therefore, we have two possible solutions: \(x = 3\) or \(x = -3\).
The solutions to the equation \(-2|x| = -6\) are \(x = 3\) and \(x = -3\). On a number line, this is represented by placing filled dots at the positions \(-3\) and \(3\). Any number line that features points specifically at \(-3\) and \(3\) is the correct representation for the solution set \(\{-3, 3\}\).
Imagine you are tracking the position of a sensor that is located at a certain distance from a central reference point. If the sensor can be at any position \(x\) such that its distance from the origin is exactly 3 meters, you would look for sensors at the positions \(x = -3\) and \(x = 3\). Understanding how to solve equations involving absolute values allows you to determine all possible locations that satisfy a given distance constraint, which is essential in fields like physics, engineering, and navigation.