Question

select the correct answer.

what is the solution to \\(\frac{1}{2}|x| = -3\\)?

\\(x = -6\\)
\\(x = -6\\) or \\(x = 6\\)
\\(x = -6\\) or \\(x = 2\\)
no solutions exist.

Explanation:

Isolate the absolute value expression

We begin with the given equation:

$$ \frac{1}{2}|x| = -3 $$

To isolate the absolute value term, multiply both sides of the equation by \(2\):

$$ |x| = -6 $$

Analyze the range of absolute value

The absolute value of any real number represents its distance from zero on the number line.
By definition, distance is always non-negative:

$$ |x| \ge 0 $$

Determine the solution set

Since \(|x|\) must be non-negative, it can never equal a negative number:

$$ |x| = -6 \quad \text{has no real solutions.} $$

Therefore, no solutions exist for this equation.

Answer:

• (A) \(x = -6\)
• (B) \(x = -6\) or \(x = 6\)
• (C) \(x = -6\) or \(x = 2\)
• (D) No solutions exist. (Correct answer)