Represent the following inequations graphically in two dimensional plane and hence solve them:
(i) `|x|lt2`
(ii) `|y|ge3`.

Answer

To solve the given inequations graphically, we will follow these steps:

### Step 1: Solve the first inequation |x| < 2

1. **Understanding the Inequation**: The expression |x| < 2 means that the distance of x from 0 is less than 2. This can be rewritten as:
\[ -2 < x < 2 \]

2. **Graphing the Inequation**: On a number line, we will plot the points -2 and 2. Since the inequality is strict (less than), we will use open circles to indicate that -2 and 2 are not included in the solution set.

3. **Shading the Region**: We shade the region between -2 and 2 to represent all the values of x that satisfy the inequality.

### Step 2: Solve the second inequation |y| ≥ 3

1. **Understanding the Inequation**: The expression |y| ≥ 3 means that the distance of y from 0 is greater than or equal to 3. This can be rewritten as:
\[ y ≤ -3 \quad \text{or} \quad y ≥ 3 \]

2. **Graphing the Inequation**: On the y-axis, we will plot the points -3 and 3. Since the inequality is inclusive (greater than or equal to), we will use closed circles to indicate that -3 and 3 are included in the solution set.

3. **Shading the Regions**: We shade the regions below -3 and above 3 to represent all the values of y that satisfy the inequality.

### Step 3: Combine the Results

1. **Final Representation**: The solution set for |x| < 2 is represented by the shaded region between -2 and 2 on the x-axis, while the solution set for |y| ≥ 3 is represented by the shaded regions below -3 and above 3 on the y-axis.

2. **Graphical Representation**: The final graphical representation will show a rectangular area where x is between -2 and 2 and y is either less than or equal to -3 or greater than or equal to 3.

### Summary of the Solution

- The solution for |x| < 2 is the interval (-2, 2).
- The solution for |y| ≥ 3 is the union of the intervals (-∞, -3] and [3, ∞).