If `(1)/(2)|x|=1 and |y|=x+1`, then `y^(2)` could be

Options

A. `2`
B. `3`
C. `4`
D. `9`

Answer

To solve the problem step by step, we will start from the given equations and derive the possible values of \( y^2 \).

### Step 1: Solve for \( x \)
We start with the equation:
\[ \frac{1}{2} |x| = 1 \]
To eliminate the fraction, we multiply both sides by 2:
\[ |x| = 2 \]
The absolute value equation \( |x| = 2 \) gives us two possible values for \( x \):
\[ x = 2 \quad \text{or} \quad x = -2 \]

### Step 2: Solve for \( y \)
Next, we use the second equation:
\[ |y| = x + 1 \]
We will consider both possible values of \( x \) from Step 1.

**Case 1:** If \( x = 2 \):
\[ |y| = 2 + 1 = 3 \]
This gives us two possible values for \( y \):
\[ y = 3 \quad \text{or} \quad y = -3 \]

**Case 2:** If \( x = -2 \):
\[ |y| = -2 + 1 = -1 \]
Since the absolute value cannot be negative, this case does not yield any valid solutions for \( y \).

### Step 3: Calculate \( y^2 \)
Now we will calculate \( y^2 \) for the valid values of \( y \) from Case 1:
\[ y = 3 \quad \Rightarrow \quad y^2 = 3^2 = 9 \]
\[ y = -3 \quad \Rightarrow \quad y^2 = (-3)^2 = 9 \]

Thus, the only possible value for \( y^2 \) is:
\[ y^2 = 9 \]

### Final Answer
The possible value of \( y^2 \) is \( 9 \).

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