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If x = -|x|, then which one of the follo...

If `x = -|x|`, then which one of the following statements could be true?
I. `x = 0`
II. `x < 0`
III. `x > 0`

A

None

B

2 only

C

3 only

D

1 and 2

Text Solution

AI Generated Solution

The correct Answer is:
To solve the equation \( x = -|x| \), we will analyze the implications of this equation based on the definition of the absolute value function. ### Step 1: Understanding the Absolute Value The absolute value \( |x| \) is defined as: - \( |x| = x \) if \( x \geq 0 \) - \( |x| = -x \) if \( x < 0 \) ### Step 2: Substitute Cases Based on the Definition We will consider two cases based on the value of \( x \). #### Case 1: \( x = 0 \) If \( x = 0 \): - The left-hand side (LHS) is \( x = 0 \). - The right-hand side (RHS) is \( -|0| = -0 = 0 \). - Since LHS = RHS, this case holds true. #### Case 2: \( x < 0 \) If \( x < 0 \): - The LHS remains \( x \). - The RHS becomes \( -|x| = -(-x) = x \). - Since LHS = RHS, this case also holds true. #### Case 3: \( x > 0 \) If \( x > 0 \): - The LHS is \( x \). - The RHS becomes \( -|x| = -x \). - Since LHS \( (x) \) cannot equal RHS \( (-x) \) for any positive \( x \), this case does not hold true. ### Conclusion From our analysis: - The statement \( x = 0 \) is true. - The statement \( x < 0 \) is true. - The statement \( x > 0 \) is false. Thus, the statements that could be true are: - I. \( x = 0 \) (True) - II. \( x < 0 \) (True) - III. \( x > 0 \) (False) ### Final Answer The correct options are I and II. ---
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