Which number line represents the solutions to $-2|x|=-6$ ?

Question

amel65 amel65

03-11-2026
Mathematics

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Which number line represents the solutions to $-2|x|=-6$ ?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T17:04:13-05:00

Isolate the absolute value term by dividing $-2|x| = -6$ by $-2$, yielding $|x| = 3$.
Recognize that $|x| = 3$ corresponds to the points $x = 3$ and $x = -3$, as these are the numbers whose distance from zero is 3.
Represent these solutions as points on a number line at $-3$ and $3$.
Conclude that the final solution set is $\boxed{-3, 3}$.

Explanation

Problem Analysis and Objective

The objective is to identify the solutions to the absolute value equation $-2|x| = -6$ and understand how these solutions are represented on a number line. An absolute value expression, denoted by $|x|$, represents the non-negative distance of a number $x$ from zero on the number line. Our strategy will be to isolate the absolute value term and then solve for $x$.

Solving the Absolute Value Equation

To solve the equation $-2|x| = -6$, we first isolate the term $|x|$. We do this by dividing both sides of the equation by $-2$:
\[\frac{-2|x|}{-2} = \frac{-6}{-2}\]
Simplifying this, we get:
\[|x| = 3\]
This equation tells us that the distance of $x$ from zero is exactly $3$. On a number line, there are two points that are exactly $3$ units away from zero: $3$ and $-3$. Thus, the solutions to the equation are $x = 3$ and $x = -3$.

Number Line Representation and Conclusion

The solutions $x = 3$ and $x = -3$ correspond to specific locations on a number line. To represent these solutions, we draw a number line and mark points at $-3$ and $3$. These two distinct points indicate the values of $x$ that satisfy the original equation $-2|x| = -6$. Hence, the correct representation is a number line with points at $-3$ and $3$. Therefore, the solutions to the equation are $\boxed{-3, 3}$.

Examples

Absolute value equations are essential in many real-world scenarios, such as measuring tolerances in manufacturing. For example, if a machine part is designed to be 10 cm long with a tolerance of 0.1 cm, the actual length $L$ must satisfy $|L - 10| \leq 0.1$. This inequality ensures that the deviation from the target length is within acceptable limits, demonstrating how absolute values help define safe operating ranges and maintain precision in engineering and design.