Question:medium

The area of the region satisfying the inequalities \(|x|-y≤1,y≥0\) and \(y≤1\) is

Updated On: Jan 15, 2026
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Solution and Explanation

The graph representing the inequalities \(|x|-y≤1,y≥0\) and \(y≤1\) is shown below:
The graph of (|x|-y<=1,y>=0) and (y<=1)

The area calculation is performed using the following decomposition: \[ \text{Area of } ABCD = \text{Area of } EFCD - \text{Area of } EAD - \text{Area of } BFC \]

Step-by-step calculation:

The area is calculated using the formula: \[ \text{Area} = EF \times FC - \frac{1}{2} \times EA \times ED - \frac{1}{2} \times BF \times FC \]

Substituting the given values: \[ = 4 \times 1 - \frac{1}{2} \times 1 \times 1 - \frac{1}{2} \times 1 \times 1 \]

Simplifying the expression: \[ = 4 - \frac{1}{2} - \frac{1}{2} = 4 - 1 = \boxed{3} \]

✅ Final Answer: \( \boxed{3} \) square units

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