Question

which number line represents the solutions to |-2x| = 4?

option 1: number line with points at -4 and 2
-10 -8 -6 -4 -2 0 2 4 6 8 10

option 2: number line with points at -2 and 4
-10 -8 -6 -4 -2 0 2 4 6 8 10

option 3: number line with points at -2 and 2
-10 -8 -6 -4 -2 0 2 4 6 8 10

option 4: number line with points at -8 and 8
-10 -8 -6 -4 -2 0 2 4 6 8 10

Explanation:

Step1: Solve the absolute value equation

The equation is \(|-2x| = 4\). By the definition of absolute value, if \(|A| = B\) (where \(B\geq0\)), then \(A = B\) or \(A=-B\). So we have two cases:
Case 1: \(-2x = 4\)
To solve for \(x\), divide both sides by \(-2\): \(x=\frac{4}{-2}=-2\)
Case 2: \(-2x=-4\)
Divide both sides by \(-2\): \(x = \frac{-4}{-2}=2\)

Step2: Identify the number line with -2 and 2

We need to find the number line that has blue dots (representing the solutions) at \(x = -2\) and \(x = 2\). Looking at the options, the third option (when we consider the order of the options) has dots at \(-2\) and \(2\). Wait, let's check the options again. Wait, the first option has dots at -4 and 2? No, wait the options:

First option: dots at -4 and 2? Wait no, the first number line: marks at -10, -8, -6, -4 (dot), -2, 0, 2 (dot), 4, 6, 8, 10. Wait no, maybe I misread. Wait the second option: dots at -2 and 4? No, third option: dots at -2 and 2? Wait no, let's re-express the equation solution. We found \(x = 2\) and \(x=-2\). So the number line should have dots at -2 and 2. Let's check the options:

Looking at the options, the third option (the one with dots at -2 and 2) is the correct one? Wait no, wait the options:

Wait the first option: dots at -4 and 2? No, first number line: -10, -8, -6, -4 (dot), -2, 0, 2 (dot), 4, 6, 8, 10. Wait no, maybe the options are:

First option: dots at -4 and 2?

Second option: dots at -2 and 4?

Third option: dots at -2 and 2?

Fourth option: dots at -8 and 8?

Wait no, the equation is \(|-2x| = 4\). Let's solve again. \(|-2x| = 4\) implies \(|2x| = 4\) (since absolute value of -2x is same as absolute value of 2x). Then \(2x = 4\) or \(2x=-4\), so \(x = 2\) or \(x=-2\). So the solutions are \(x = 2\) and \(x=-2\). So the number line with dots at -2 and 2 is the correct one. Looking at the options, the third option (the one where the dots are at -2 and 2) is the correct choice. Wait, let's check the options again as per the user's image:

The options are:

1. Dots at -4 and 2

1. Dots at -2 and 4

1. Dots at -2 and 2

1. Dots at -8 and 8

Wait no, maybe the user's options are:

First number line: dots at -4 and 2

Second: dots at -2 and 4

Third: dots at -2 and 2

Fourth: dots at -8 and 8

So the correct one is the third option (dots at -2 and 2) because the solutions are \(x = 2\) and \(x=-2\).

Answer:

The number line with blue dots at \(-2\) and \(2\) (the third option among the given choices, typically represented as the option with dots at \(-2\) and \(2\) on the number line).