Number of solution pair of equations `y=||x|-2|-2| and y=(x+2)/2`equals to

Options

A. `1`
B. `2`
C. `3`
D. `4`

Answer

To find the number of solution pairs of the equations \( y = ||x| - 2| - 2 \) and \( y = \frac{x + 2}{2} \), we can follow these steps:

### Step 1: Analyze the first equation \( y = ||x| - 2| - 2 \)

1. **Break down the absolute values**:
- The expression \( ||x| - 2| \) can be analyzed in two cases based on the value of \( |x| \):
- Case 1: \( |x| \geq 2 \) → \( ||x| - 2| = |x| - 2 \)
- Case 2: \( |x| < 2 \) → \( ||x| - 2| = 2 - |x| \)

2. **Substituting into the equation**:
- For Case 1: \( y = (|x| - 2) - 2 = |x| - 4 \)
- For Case 2: \( y = (2 - |x|) - 2 = -|x| \)

### Step 2: Determine the piecewise function

The piecewise function can be defined as:
\[ y = \begin{cases} |x| - 4 & \text{if } |x| \geq 2 \\ -|x| & \text{if } |x| < 2 \end{cases} \]

### Step 3: Graph the first equation

1. **For \( |x| \geq 2 \)**:
- The graph of \( y = |x| - 4 \) is a V-shaped graph that opens upwards starting from the point \( (2, -2) \) and \( (-2, -2) \).

2. **For \( |x| < 2 \)**:
- The graph of \( y = -|x| \) is a V-shaped graph that opens downwards, intersecting the y-axis at \( (0, 0) \).

### Step 4: Analyze the second equation \( y = \frac{x + 2}{2} \)

1. **Rewrite the equation**:
- This is a linear equation with a slope of \( \frac{1}{2} \) and a y-intercept of \( 1 \).

### Step 5: Find the intersection points

1. **Set the two equations equal to each other**:
- For \( |x| < 2 \):
\[ -|x| = \frac{x + 2}{2} \]
- For \( |x| \geq 2 \):
\[ |x| - 4 = \frac{x + 2}{2} \]

2. **Solve for intersections**:
- **For \( |x| < 2 \)**:
- \( -x = \frac{x + 2}{2} \) (for \( x < 0 \))
- Solve to find intersection points.
- **For \( |x| \geq 2 \)**:
- \( x - 4 = \frac{x + 2}{2} \) (for \( x \geq 0 \))
- Solve to find intersection points.

### Step 6: Count the number of solutions

1. **Graph both functions**:
- The graph of \( y = ||x| - 2| - 2 \) will intersect the line \( y = \frac{x + 2}{2} \) at specific points.
- By observing the graphs, we can determine the number of intersection points.

### Conclusion

After analyzing both equations and their graphs, we find that there is **one intersection point** between the two graphs.

### Final Answer
The number of solution pairs of the equations is **1**.

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