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Chapter 0: Problem 111

Let \(R\) be the region consisting of the points \((x, y)\) of the Cartesian plane satisfying both \(|x|-|y| \leq 1\) and \(|y| \leq 1 .\) Sketch the region \(R\) and find its area.

Short Answer

Expert verified
The region \(R\) is a diamond shape centered at the origin with corners at (1, 0), (0, 1), (-1, 0), and (0, -1). Its area is 4 square units.

Step by step solution

01

Sketching the Functions

The first inequality is a transformation of the absolute value function \( |x| \), and similarly the second inequality is a transformation of the absolute value function \( |y| \). Sketch these functions by graphing the line \( y = x \) for \( x \leq 0 \) and \( y = -x \) for \( x \geq 0 \) . Then shade the region where both inequalities are satisfied.
02

Identifying the Region

The resulting graph from step 1 is a diamond shape centered at the origin with corners at (1, 0), (0, 1 ), (-1, 0), and (0, -1). The inequality \( |y| \leq 1 \) is satisfied in the bounded region between the horizontal lines \(y=1\) and \(y=-1\). Therefore, the included region \(R\) is a diamond shape.
03

Calculating the Area

Since the shape is a square, rotated by 45 degrees, with sides of 2 units length, its area can be calculated using the formula for the area of a square \(Area = side^2\).
04

Final Calculation

Substitute the given side length into the formula. So the area is \(2^2 = 4\) square units.

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