Graph the inequalities \( y \geq|x-2| \) and \( y \leq 4-|x| \). What is the area of the enclosed region? Homework Help

To graph the inequalities \( y \geq |x-2| \) and \( y \leq 4 - |x| \), start with the equations of their boundaries. The first, \( y = |x-2| \), creates a V-shape vertex at (2,0) and opens upwards. The second, \( y = 4 - |x| \), is an inverted V, peaking at (0, 4) and sloping downwards. The area of the enclosed region can be found by identifying the points where these two graphs intersect. Setting \( |x-2| = 4 - |x| \) leads to calculating solutions in various scenarios based on the definitions of absolute values. You’ll find that the points of intersection occur at \( (0, 4) \) and \( (4, 0) \). The enclosed shape forms a quadrilateral when you connect these points with the lines of the inequalities clearly defined. The area can be calculated using trapezoidal formulas or by integrating the boundaries. In this case, the area comes out to be 8 square units.