Conside the curves
`y=|x-1|` and `|x|=2`
What is /are the point(s) of interesection of the curves?

Options

A. A. only (-2,3)
B. B. only (2,1)
C. C. (-2,3) and(2,1)
D. D. Neither (-2,3) nor (2,1)

Answer

To find the points of intersection of the curves \( y = |x - 1| \) and \( |x| = 2 \), we will follow these steps:

### Step 1: Understand the equations
The first equation is \( y = |x - 1| \). This represents a V-shaped graph that opens upwards with its vertex at the point (1, 0).

The second equation \( |x| = 2 \) indicates that \( x \) can take two values: \( x = 2 \) and \( x = -2 \).

### Step 2: Substitute the values of \( x \)
We will substitute \( x = 2 \) and \( x = -2 \) into the first equation to find the corresponding \( y \) values.

#### For \( x = 2 \):
\[ y = |2 - 1| = |1| = 1 \]
Thus, one point of intersection is \( (2, 1) \).

#### For \( x = -2 \):
\[ y = |-2 - 1| = |-3| = 3 \]
Thus, another point of intersection is \( (-2, 3) \).

### Step 3: Summarize the points of intersection
The points of intersection of the curves \( y = |x - 1| \) and \( |x| = 2 \) are:
1. \( (2, 1) \)
2. \( (-2, 3) \)

### Final Answer:
The points of intersection are \( (-2, 3) \) and \( (2, 1) \).

---